Breaks a black hole in time and space

Black holes: the singularity of the previous day

Black holes are singularities of general relativity. You are shielded from the outside world by an event horizon. Good thing because behind it the time could possibly tick both towards the future and towards the past

Long before chemists and physicists were able to image individual molecules or atoms with X-rays or scanning tunneling microscopes, they developed the atomistic theory of matter. Up until the beginning of the twentieth century it was a fairly plausible assumption that all substances consist of atoms: all experiments pointed in this direction. For example, Albert Einstein could be in 1905 annus mirabiliswho interpret the chaotic Brownian movement of microscopic particles in liquids as a result of the accidental repulsion of invisible atoms. But that was one indirect Review of the atomistic theory.

The situation is similar with the black holes in space. A black hole is a gravitational giant that inevitably devours everything, i.e. both matter and radiation, so that we cannot look inside from the outside. Photons that cross the event horizon barrier can no longer escape the attraction of the black hole, i.e. physicists can only use the formula machinery of general relativity to calculate what probably happens in the belly of the black hole. So one tries to understand, again indirectly, what is going on inside.

This is how science works: We examine the smallest and the largest simply "because it's there" (like Mount Everest), because we want to reduce our ignorance about the world step by step. If nineteenth-century chemists and physicists had not acted like this, they would never have dared to study atoms, because at that time they were beyond human sight. The imagination, paired with appropriate mathematical formulations, is, however, the best microscope in science. Hypotheses are set up, which are then tested experimentally. With many conjectures in physics we do not immediately know how to test them experimentally, but it is almost always only a matter of time before someone suggests a suitable experiment and it is carried out.

Black holes are of course a theoretical construct of modern physics that pushes our imagination and today's theory of gravity to its limits. What happens inside black holes strains "healthy" intuition, as do the paradoxes of quantum mechanics.

It is therefore surprising that the possibility of something like black holes was discussed as early as the 18th century (under the name "dark stars"). John Michell in England and Pierre-Simon de Laplace in France thought about massive stars almost at the same time. Since the escape speed of an object from the gravity of a celestial body increases in direct proportion to its radius (with constant mass density), it was conceivable that even light could not reach the necessary escape speed for very large stars and thus remain "trapped" in the star.

Michell thought that such "dark stars" could be detected if a bright and a dark star rotated as a pair around the common center of gravity. Astronomers could observe the orbit of the bright star and in this way infer the existence of the dark companion.1 However, the same basic idea can also be pursued in another direction, which neither Michell nor Laplace took into account: If a spherical mass is so strongly compressed, that a critical mass to radius ratio is reached, the escape speed can exceed the speed of light. We also get such a black hole, and therefore there should be black holes in all variations: smaller than atoms, but also larger than our solar system.

The Schwarzschild radius

When we talk about the size of a black hole, we usually talk about the amount of its "Schwarzschild radius". Gravitation only has an attractive effect and so matter can clump together in asteroids, planets or stars. Stars are the chemical laboratories of the universe: Driven by nuclear fusion, they "cook" from hydrogen and helium, and also in collisions, almost every other element in the periodic table. The real gravitational giants, however, are the black holes, which can soak up tens or millions of solar masses.

As mentioned above, black holes can even be understood using Newtonian mechanics. For a heavenly body of the earth M. and radius R. is the escape speed v given by the following expression:

v = √(2GM/R.)

where G is the universal gravitational constant. The formula calculates the minimum speed that a projectile must have in order to be able to escape from the gravitational field of the celestial body. If the mass M increases (with R constant) or the radius decreases R. (With M. constant), can v get bigger and bigger until the speed of light is reached. Since no projectile can fly away faster than this, nothing can escape from such a "black hole". The critical radius for this (if v= c), derived from the above equation, is:

R. = 2GM./ c2

This is called the "Schwarzschild radius", in honor of Karl Schwarzschild, who, within the framework of general relativity, proposed the solution to the Einstein equations that now bears his name. Schwarzschild also calculated the critical radius for what were then known as the "Schwarzschild singularities". An example: if you enclose a solar mass in a sphere with a radius of 3 km, the result is a black hole. This is how so-called stellar black holes are actually formed, i.e. by the collapse of matter from extinct stars to the center of mass. When the nuclear fusion inside a star can no longer stop the gravitational pressure of the star's mass, this mass collapses and forms a black hole.

Einstein and the black holes

Today, when new black hole sightings are reported in the universe every day, it is difficult to believe that Einstein was still doubting the reality of the "Schwarzschild singularities" in 1939 (it was not until around 1963 that the singularities were called "black holes" 2). In an article for a mathematical journal3, Einstein wrote:

The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality (...) The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.

However, just a few months later, Robert Oppenheimer and Hartland Schneider were able to use Einstein's theory of relativity to derive the process of black hole formation from the equations of the theory

How to fall into a black hole

Newton's theory of gravity is simple: the force with which two masses attract is proportional to the product of the masses and inversely proportional to the square of the distance. The attraction is transmitted instantaneously, as a kind of action at a distance. However, this was a flaw in the theory, with which Newton was never satisfied.

Einstein's theory, on the other hand, is a local theory. She explains how masses deform space and time around them. Instead of an effect at a distance, we have a local effect on space and time. Einstein's tensor formula is reduced to ten so-called differential equations, which explain how the next part of the space-time continuum is calculated from given initial conditions, and from this the next, etc. We can therefore use the computer for a given mass distribution and under certain conditions to calculate the Reconstruct the geometry of the room step by step. One condition is that no signal can travel faster than light.

Space-time diagrams can be used to represent the movement of particles, as in Fig. 3. Here there is only one spatial direction (x), while the time direction is displayed vertically. The physicists change the units of space and time so that the speed of light can be written as c = 1. Measured in such units, a light beam leaving the origin of the system can only fly away over the diagonal of the diagram (in both directions). Every other particle cannot move as fast and its trajectory remains trapped within the "light cone". Time is only ticking towards the future and allowed trajectories must take this into account.

Now if you imagine that the world has only two dimensions, we can use the vertical axis to represent time. Fig. 4 shows the future or past cone of an event at the origin of the coordinate system. In the future cone, all places are in space-time that can be influenced by the event at the origin, i.e. there are all events in space-time that can be reached by a signal from the origin (and the signal can have at most the speed of light). In the past cone are all events that could have influenced the present at the origin (through a signal). When you then move in the world, you carry your coordinate system with you. The strange thing about the theory of relativity, however, is that time does not tick globally, but that each observer carries his own clock and time with him and the time axes of observers in relative motion are not parallel to each other.

So when you're in space on a rocket, you carry your local watch with you. The past and future cones are defined locally. Something strange happens when a missile falls into a black hole (Fig. 5).

On the left side of Fig. 5 you can see an observer falling straight ahead into a black hole. The Schwarzschild radius is displayed, this is the "point of no return". Once a particle has crossed the event horizon, it can no longer get out.

On the right side of Fig. 5 you can see the corresponding space-time diagram. The time is represented by the vertical coordinate. The observer in green stays outside in the space ship (i.e. the world line only moves in time, i.e. upwards). The blue observer steers towards the black hole. The local time axis is twisted with respect to the green observer. Arrived at the event horizon, the cone of light has twisted further. Now a photon at the edge of the light cone can no longer fly away from the black hole. The photon stops exactly at the event horizon (that is why the right diagonal of the light cone is now vertical, i.e. time is running, but a light signal from the blue observer falling in the direction of the green observer never reaches its destination). In other words, the outside observer would see the space traveler's clock slowing down and stopping at the event horizon (the clock could be visualized with a light signal reflecting back and forth).

The light cone has now "overturned" inside the black hole. Oddly enough, the radial direction now becomes "time-like" while the original time axis becomes "space-like". This means that radial movement behind the event horizon is only allowed if the observer is incessantly striving towards the center. But as far as time is concerned, you can now head for both directions (past and future).

In other words: In the Einstein equations there is a so-called "line element" from which the geometry of space and time is reconstructed. In Karl Schwarzschild's solution, the line element is given by a combination of changes in time and changes in space. The coefficient for the time variable is

(rs/r-1)

and for the radial distance from the center of the black hole is the coefficient

(rs/r-1)-1

Outside the black hole, the radius r is larger than the Schwarzschild radius rs and rs/r is therefore less than 1. The coefficient of the time variable is therefore negative, and for the radial direction the coefficient is positive. The line element is negative overall in the Schwarzschild solution. Since only time makes a negative contribution, the change in time must not be zero. In other words, time must never stand still outside a black hole. Not even when Faust calls out to the moment: "Stay a while! You are so beautiful!".

But when the observer crosses the event horizon, r becomes smaller than rsi.e. rs/r becomes greater than 1 and the signs of the time and radial variables change!

The consequence of the change in sign is that outside the event horizon time can only tick forwards, while we can move back and forth in the radial direction to the black hole (with rocket propulsion). However, and because of the change in sign, we are allowed to move freely in time within the event horizon, but not in the radial direction. We always have to strive in the direction of the singularity, a particle falling there must never stop on the way there.

Outside the event horizon, there is a technology (rocket propulsion) that will allow us to fly away from the black hole, but not into the past. Within the event horizon we cannot stop on the way to the singularity, no matter how much energy we want to expend. But strangely enough, time travel is allowed, whereby the necessary technology (time drives?) Has yet to be invented. This is what some physicists mean when they talk about the possibility of time travel within black holes (and science fiction writers have exploited the idea often enough).

But if such time travel were possible, you could imagine it like this: If you drive on the Berlin-Singularität autobahn (this is 100 km long and Berlin is at km 0), at km 50 I can go back in time like Dr. Who travel and "land" on the autobahn again, but only beyond km 50, e.g. at km 75, i.e. always closer to the singularity. In particular, I cannot go back in time to kill myself (since I only drove normally up to 50 km).

Through time travel, however, movement with faster than light speed in the radial direction would be possible, which brings us to all paradoxes of violated physical causality. To prevent this, Roger Penrose has postulated the hypothesis of a "cosmic censorship": There should be no "naked singularities" in the universe, ie the phenomena described above would be hidden from external observers by the curtain of the event horizon, so that we are outside blacks Holes don't have a causality problem.

In "Islands of the previous day" by Umberto Eco, a stranded sailor doubts the reality of an island across the meridian of the date line in the Pacific, which separates one calendar day from the next. For the sailor yesterday lies beyond the date line, and that is why he thinks that perhaps the island he was observing is no longer there (it is the island of the "previous day"). He doesn't dare to swim there and drowns.

The Autobahn Berlin singularity would also be something like that. You can go to ruin normally or with a time machine. Right at the front, right on the edge, could be yesterday.

Rotating black holes

To relax the reader, who may already be considering joining the troll guerrilla, let me tell you that all of these statements are of course extremely speculative. The physicists are actually debating intensely whether Einstein's differential equations can still be trusted under such extreme conditions or whether another theory is necessary that can cope with singularities or avoid them altogether.

But first of all, it will get worse ... The Schwarzschild solution does not apply to rotating black holes. But when a rotating star collapses, the torque must be maintained and therefore there should also be rotating black holes. For that you need a different solution than that of Schwarzschild. It was not until 1963 that Roy Kerr presented his result for rotating and uncharged black holes.

The Kerr solution for rotating black holes has much more internal structure than the Schwarzschild solution. There is an external event horizon where the spherical shape no longer exists and the surface looks more like an ellipsoid. In addition to the external event horizon, there is also an internal event horizon, where the light cone of the falling observer is twisted again. In other words, the time and the radial component change sign for the second time and the time axis is now oriented as it is outside the black hole. Causality as we know it is being restored. This does not help the falling observer, since the singularity in the center is no longer point-shaped, but ring-shaped. Every particle is attracted there. The singularity must be ring-shaped because the torque cannot be accommodated in a point-shaped singularity.

Fig. 6 shows the structure of such a rotating black hole with two event horizons and two so-called ergoregions.The external ergonomic region corresponds to an envelope around the event horizon, within which space and time are carried along by the rotation of the black hole. This is called "frame dragging". An object that is close enough to the black hole is set in rotation because the space around the black hole itself is rotated with it. The inner event horizon also has such an area in which "frame dragging" takes place, which however is shifted towards the center of the black hole. The computation of all these areas is extremely complicated, but the result shows that a rotating black hole is a fairly structured object.

As far as time travel is concerned, the problem is that there are closed geodesics in space and time in such black holes. That means you start the journey somewhere and arrive back at the same place, but in the past. In the inner area of ​​the black hole (behind the second event horizon), this does not happen, but you cannot save yourself there because you will be repelled in the direction of the ring singularity

Gravitation theorists who want to look inside black holes do not stop there. Another possibility to look there is to map the so-called invariants of the solutions of the Einstein equations with the computer. There are several such invariants. Fig. 7 shows one of them, the so-called Weyl curvature invariant, as calculated by astrophysicists. This does not mean that these surfaces exist inside black holes, but that the combination of many of these invariants tells us the geometry at every point.

Measurement of the properties of black holes

It seems like a hopeless endeavor to be able to confirm the speculations of the theory through astronomical measurements. The mass of a black hole can, however, be determined indirectly via the effect on the orbits of stars rotating around it (see Fig. 1). The torque can perhaps also be calculated by measuring the disturbances from the same trajectories (to observe "frame dragging"). Other properties could presumably be derived from gravitational waves detected on Earth.

Gravitational wave detectors such as LIGO in the USA and VIRGO in Italy do not simply measure the deformations of the equipment caused by gravitational waves. The signal-to-noise ratio is too unfavorable for this. These laboratories search for wave patterns of black hole collisions that have been theoretically pre-calculated. This is the only way to make the suggested signals stand out from the noisy data using correlation calculations. It's a needle in a haystack, but you have a theoretical model of the needle that makes the search possible in the first place. A group of astrophysicists recently proposed to calculate the shape of these gravitational waves more precisely and to consider possible echoes.

As you can see, all of these theoretical calculations are, in part, rather speculative. That is why people have been talking about a quantum theory of gravity for decades, which possibly brings together quantum and gravity effects. Perhaps this will make the singularities and paradoxes disappear, which otherwise cannot be erased with normal instruments.

So black holes are gravitational monsters, but incredibly rich constructs that modern physics has to grit its teeth on on the way to new shores.

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