THE UNSTOPPABLE MOVEMENT OF SPACE-TIME

One of the greatest conceptual shifts in human history is the discovery that space is not a fixed, unchanging "stage" where things happen, but that, over time, it can be "stretched" or "compressed" like a rubber band.

Another characteristic of this four-dimensional structure we call space-time is that it can rotate, giving rise to a series of fascinating physical phenomena. In this article, we'll take a journey to one of the strangest and most unknown places in the Universe, a place where everything flows without stopping, where rest is impossible, where space-time itself spins endlessly, dragging everything in its path.

Spatial distances and temporal distances

The first thing we must remember is that our Universe has four dimensions: three spatial dimensions and one temporal dimension. Therefore, to measure the distance between two events A and B, we need to specify four coordinates. To understand in detail how distances are measured in space-time, let's separate spatial and temporal distances:

Spatial distances and the Euclidean metric

In flat spaces the distance between 2 points P(x1,x2,x3) and Q(x2,y2,z2) is obtained by applying the Pythagorean theorem in 3 dimensions: |PQ|=(x2-x1)2+(y2-y2)2+(z2-z1)2, if we take very small distances or differentials we have that the flat metric is simply: ds2=+dx2+dy2+dz2.

Space-time distances and the Minkosky metric

The previous method of measuring distances lacks two fundamental ingredients: the dimension of time and taking into account the absolute limit of the speed of light. Special relativity tells us that no matter the speed at which we move, all observers will always measure the same speed of light c. This means that if we emit a pulse of light at time t0, this pulse of light will move away from us at speed c, traveling a distance ct, regardless of whether we remain at rest or accelerate toward it, trying to reach it. Therefore, any observer will behave as if they were at the center of a sphere of radius ct:

In both the stationary reference frame S and the moving frame S', the sphere of light moves away at speed ct. Therefore, every observer will feel themselves at the center of a sphere of radius ct.

This implies that in any reference system the distance (the metric) S coincides with the equation of a sphere: X2+Y2+Z2=(ct)2 Therefore the relativistic 4-dimensional distance is measured as:

ds2= dX2+dY2+dZ2-d(ct)2

Now we must realize something fundamental: the temporal dimension contributes to the distance with a negative sign. Since all bodies move at a speed less than or equal to ac, the metric will always be negative or zero. If the metric is positive, it means that the measured events are outside the circle, that is, outside the light cone (there is no causality). The fundamental idea of all this is the following: the spatial dimensions, through which we can move freely, are those that have a positive sign in the metric; the temporal dimensions, which flow inexorably in a single direction, are those that have a negative sign in the metric.

The Kerr metric

The vast majority of real black holes, formed from stellar collapse, are rotating black holes, also called Kerr black holes. The metric for a rotating black hole is as follows:

where the denominators are given by:

This formula may seem intimidating, but as we'll see, we can choose only the parameters we need and explain their meaning step by step to get a general idea of what a rotating black hole really means. The parameters of the Kerr metric are:

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

- Physical magnitudes and constants: G=Newton's constant M=mass of the black hole a=constant that measures the rotation speed of the black hole

The first thing we'll do is find the points where the metric diverges. To do this, we simply look for the values of r for which the denominator (delta or rho) tends to zero. The values of r for which delta vanishes are obtained by solving the second-degree equation delta(r):

The values for which rho vanishes are: r=0 and theta=·/2 . We then have two values of r for which the metric tends to infinity (we'll ignore the last two for now): r+ and r-. We'll see what happens at these points where the distances seem to tend to infinity.

The r+ and r- horizons: when space and time exchange roles

Let’s consider the motion of a body moving only in the radial dimension r. The coefficient of the metric dr2 is: rho2/delta(r) As we saw earlier, the square roots of delta(r) are r+ and r- which means that the parabola of this second degree equation cuts the x=0 axis at r+ and r-. This means that this function has positive values for values of r greater than r+ and negative for values of r less than r+, that is, the coefficient of the metric is positive as corresponds to a spatial dimension for values greater than r+, but once it passes through r+ the metric becomes negative! Something incredible happens at this point: the spatial dimension r behaves like a temporal dimension. This means that the dimension r through which we could previously move freely now flows irremediably forward as if it were a temporal dimension. Space and time have exchanged roles!

No matter the speed at which we move, no matter the direction we take, the radial coordinate will move in the direction of decreasing r (toward the singularity). This is the real reason why general relativity predicts the existence of event horizons and therefore black holes: once r+ is crossed, nothing can escape back outward; the motion of spacetime itself prevents it. Therefore, r+ is the event horizon of the black hole (r- constitutes a second, internal event horizon).

Ergosphere: a world where there is no rest

No physical object in this Universe will ever be able to carry direct information about the phenomenon we saw in the previous section, since once the r+ boundary is crossed, nothing can escape. However, in rotating black holes, there is another "strange" boundary outside of r+, a region we can enter and exit again, a region with properties as extraordinary as, or even more so than, the event horizon. It is now that we enter one of the strangest worlds in the Universe: the ergosphere. We have seen that upon crossing r+, the coefficient of the metric r changes from positive to negative. If we analyze when this happens in the case of the coefficient of the time metric, we find something astonishing: the coefficient of dt2 is:

The above expression becomes positive when:

The roots of the previous equation (the points where it is cancelled and therefore where the parabola cuts the x=0 axis) are: GM+-· (G2M2-a2cos2fi)

If we compare this value with the position of the event horizon, we find that the coefficient of the time dimension becomes positive before reaching the r+ horizon! This means that from this point on, time behaves as if it were a spatial dimension. What does this mean?

The four zones of a rotating black hole

To see what happens when we cross the area known as the "Killing horizon" and enter the ergosphere, let's analyze the trajectory of a photon traveling radially toward the black hole in the direction of the equatorial plane. In the Kerr metric, we take ds2 = 0 (for a photon, the metric is zero), fi = /2 (equatorial plane), 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:

The two expressions above show us something amazing: right at the edge of the ergosphere a photon only has two options:

- Move in the opposite direction to the hole's rotation (first expression). In this case, its instantaneous velocity is zero.

- Move in the direction of rotation of the hole (second expression).

In this case , its speed matches the black hole's spin rate . Upon entering the ergosphere, the time coordinate is now positive, so nothing can move in the opposite direction to the black hole's spin. From this, it follows that no body can be at rest; all bodies must move in the direction of the hole's spin . The black hole drags 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 spin of the black hole behaves like a coordinate that marks the passage of time.

Objects with negative energy

In the previous section, we saw the first strange physical phenomenon within the ergosphere: rest does not exist within it. Now we will see another, no less strange, phenomenon: objects with negative energy. In general relativity, the energy of a body is measured by a mathematical object called a tensor. Under "normal" conditions, it is always possible to define an inertial reference frame (absent external forces) in which to define this energy tensor. However, in a rotating black hole, this inertial reference frame does not exist (everything is in rotational motion), so the usual measurement of an object's energy is not valid. We know from Noether's theorem that energy is the conservation of a magnitude over time ; however, as we have seen, the time dimension in the ergosphere goes from having a negative sign to a positive sign, which opens the door to a truly unusual phenomenon: for an observer at rest far from the black hole, an object in the ergosphere can have, under certain circumstances, negative energy . It should be noted that negative energy would be measured by an observer located very far from the black hole and that an object with negative energy will always follow a trajectory that will never leave the black hole, so it will never be measured by an outside observer. To achieve negative energy for an object within the ergosphere, we must apply negative angular momentum to it; that is, the object must rotate in the opposite direction to the black hole's rotation (intuitively, we can begin to glimpse why this object has negative energy).

Let's imagine that we then throw this object into the black hole. Inside the ergosphere the escape velocity is less than c since we are outside the event horizon, in fact there are trajectories (geodesics) in which there is a point (called the return point) in which the velocity is cancelled allowing the object to change direction and return to the outside. When the object P1 that we have thrown reaches the return point we make the object split in two so that the piece with negative energy P2 heads inwards and the piece with positive energy P3 heads outwards. By the principle of conservation of energy we have: P1 = -P2 + P3. Therefore P3 = P1 + P2 The object (P3) will leave the black hole with a greater energy than it entered! This phenomenon called the Penrose cycle implies that it is possible to extract energy from a rotating black hole, this energy comes from the rotational energy of the same.

There are calculations that indicate that a huge percentage of the available energy in the Universe is stored in the form of rotational energy inside black holes. Closed time loops In the first section of this article we saw that the metric tends to infinity in two cases: delta=0 or rho=0 but we only analyzed the first case. The second case rho=0 is fulfilled when r=0 and also fi=·/2. This condition is not fulfilled at a single point but at all points of the "circle" fi=·/2 (it is actually a ring) . The central singularity is not a point but a ring! As if this were not already "strange" enough, Mathematics tells us that, in principle, it is possible to pass through the ring, which theoretically would lead us to access a different space-time inside a different black hole (white hole).

We've saved perhaps the most unusual phenomenon for last: the ring is a "time-like" surface, meaning its trajectory runs through the temporal dimension (which, due to its enormous curvature, closes in on itself). If we stand right next to the ring and follow a closed trajectory around it, we'll travel along the temporal dimension, so we'd return to the starting point—back to the past! Now that would truly be time travel to the past!

Conclusions and final observations

Space-time is a four-dimensional entity capable of compressing, stretching, or rotating, which produces truly incredible phenomena. It should be noted that, at present, it is fair to say that no one knows exactly what happens when we cross the event horizon (this article is based exclusively on the view of general relativity and ignores possible quantum effects). Since we do not have a quantum theory of gravity, we cannot yet calculate what happens when both theories must be taken into account. In fact, there are indications that in the Kerr metric, the region of space-time interior to r- is unstable. Despite this, the phenomena described in the ergosphere are, in principle, real (in this region, we can ignore possible quantum effects). Therefore, we can say that the ergosphere is probably the strangest and most fascinating region of the accessible Universe that we can, in principle, visit.

Sources:Geodesic of Kerr metric , The Kerr space time: a brief introduction