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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Dong Weiyuan

Schwarzschild black hole, RN black hole, Kerr black hole, Kerr-Newman black hole. The cave is wonderful.

Black holes are frequent visitors in popular science content, such as "singularity of time and space", "event horizon" and "Schwarzschild radius". These nouns have long been familiar to enthusiasts. But when it comes to the multi-layered structure inside a black hole, I'm afraid many people will be baffled. There is no matter in a black hole, only a seriously distorted space-time. How can it have a hierarchical structure like an egg? In fact, black holes not only have internal structures but also are very complex, but we can start with "0".

In fact, the black hole that often appears in popular science books is only the simplest kind in the family of black holes, which is called Schwarzschild black hole. This kind of black hole has neither electricity nor rotation, but only one physical property-mass. Under the premise of such a high degree of simplification and isotropic symmetry, of course, there is no chance of too complex structure. But in the real universe, most celestial bodies have rotational angular momentum and more or less have some electric charge, and black holes should be no exception. When rotational angular momentum and electric charge are added to the theoretical model describing black holes, some interesting structures appear.

The classification of black holes under general relativity we all know that the structure of a Schwarzschild black hole is a sphere called the event horizon, which wraps the singularity of space-time at the center of the sphere, and this part of the region from the event horizon to the singularity is an irreversible one-way region. anything that falls into this area will inevitably go to the singularity. There is a full sense of gimmick: in this one-way area, time becomes space, space becomes time. As for how to understand this sentence, we will talk about it later.

Now we let the black hole carry an electric charge, that is, the RN black hole, it has internal and external horizon, the unidirectional region only exists between the two horizon, the more charge the black hole carries, the thinner the unidirectional region of the spherical shell. On the other hand, the region within the inner event horizon returns to the appearance of ordinary space-time, and there is no exchange of time dimension and space dimension, and the singularity of the center of the black hole lies in this ordinary space-time region.

If the black hole has rotation, that is, the Kerr black hole, its event horizon is no longer a symmetrical sphere, but a pumpkin-like surface, and the pumpkin skin-like event horizon also has two layers, with an one-way zone in the middle. In addition, the Kerr black hole has two more interfaces than the RN black hole-- the outer static surface and the inner static surface-- outside the outer event horizon and inside the inner horizon, respectively. The region from the rest surface to the event horizon is called the energy layer, and the name comes from Penrose's discovery that energy can be obtained from this area. The most interesting part of the Kerr black hole is that there is no longer a singularity in the center and is replaced by a strange ring.

The structure of the Kerr black hole has basically reached the limit of complexity. The structure of the Kerr-Newman black hole with electric charge is not more complex than that of the Kerr black hole, and it still looks like an one-way region sandwiched between the internal and external energy layers. in the middle is still an odd ring that represents the singularity of space-time. The amount of charge only adds one more parameter to the specific location of these structures.

At this point in the Schwarzschild metric, we have taken a quick look at the structure of four kinds of black holes, but I am sure that most readers will not be satisfied with such a general view. In order to speak more clearly, we first take half a minute to understand two physical concepts in the theory of relativity-"line element" and "metric".

The "line element" can be roughly understood as the tiny interval between two adjacent points in time and space, recorded as ds. In the air when it is straight

Or in the form of polar coordinates.

What it looks like as a vector inner product is

The 4 × 4 matrix sandwiched in the middle is the metric, which shows the geometric properties of space-time. The metric of flat space-time is a simple diag (- 1, 1, 1, 1) diagonal matrix, while the metric of curved space-time becomes more complicated.

The so-called solution of the general relativity equation is to calculate all the components of the metric. All the depictions of the geometric properties of space-time are hidden in this matrix.

Knowing this, we can read the space-time metric according to the expression of a line element, and then guess what space-time looks like. For example, if an object with no rotation, no charge and mass M is placed at the origin of polar coordinates, the expression of the vacuum line element around it is

Among them

We can immediately see that this Schwarzschild metric is still a diagonal matrix, but compared with the appearance of flat space-time, the blue components of gtt and grr are obviously different. These two terms are the starting point for the study of black holes in the post-relativistic era. Rs is the Schwarzschild radius and r=rs is the event horizon of the Schwarzschild black hole.

When s →∞, the Schwarzschild metric returns to the appearance of flat space-time, indicating that the effect of space-time bending at infinity gradually disappears. So how does space-time bend near a black hole? Let's send an adventurer to inspect near the event horizon. In three-dimensional space, the position of the adventurer is a point, while in four-dimensional space-time, due to the passage of time, even if the adventurer is still motionless, this position is still a line, which is called "world line".

The theory of time expansion relativity near the black hole tells us that the world line is an absolute physical object, and no matter which frame of reference it is calculated, the length of the same ds on this line must be the same. We choose two special frames of reference, one is the reference frame of relative black hole rest, and the other is the adventurer's own follow-up frame of reference.

In the previous frame of reference, we still use the formula already mentioned to calculate the line elements of the adventurer's world line.

Among them, it represents the change of the tangential position along the sphere, together with the radial position change dr, gives the change of the spatial coordinate position of the adventurer. What needs to be noted is the dt in the equation, which represents the time change felt by the viewer standing at infinity and relatively still in the black hole.

In the latter frame of reference, the adventurer himself does not change any position relative to the follow-up frame of reference, but simply experiences the passage of time, so the line element is simplified to

Where d τ is the time change felt by the adventurer himself.

The world line of adventurers in the two frames of reference is the same, so

Now, we order the adventurer to hover, so dr and d Ω are both 0, and the formula is simplified to

If the hovering position satisfies the r=1.01rs, there will be a dt ≈ 10d τ, and the adventurer's clock becomes so slow! Viewers at infinity have to wait a decade to see that the adventurer has grown one year old. this is the time expansion effect of the gravitational field. If the adventurer emits a beam of light into the distance, by the time the light reaches the distant viewer, the frequency has dropped to 1/10 of what it was at the time of departure, that is, a serious redshift.

When the position of the adventurer is infinitely close to rs, the time expansion is also infinitely close to infinity. Although the adventurer himself is still experiencing the normal passage of time, from a distant point of view, the adventurer's time has almost stopped, and the frequency of its light is infinitely close to zero. In other words, light from the event horizon cannot transmit energy to the distance. It suddenly makes people feel that there are so many TV programs and even radio stations that are willing to take the "horizon" as their name.

The space dimension becomes the time dimension. Back to physics, let's see what happens after the adventurer crosses the event horizon and enters the interior of the black hole. Some people may question: the adventurer has reached the limit of time expansion at the event horizon, and the distant viewer cannot wait for the adventurer to cross the horizon even when the universe is destroyed.

Smaller, smaller pattern.

Although the distant viewer cannot wait in the time he or she has experienced, it does not mean that the adventurer cannot arrive. In fact, according to the time experienced by the adventurer himself, he can reach and cross the event horizon in a limited time. Of course, it would be better for him to have very hard armor and a very small body so as not to be torn apart by tidal forces.

When r < rs, the adventurer enters the one-way zone. Let's take a look at how time and space are interchanged here.

At this time

It still stands.

In order for the adventurer to feel that the time d τ is a real number, the left side of the equal sign must also be negative, and when r < rs

Can only rely on

The contribution of this project.

In other words, after entering the event horizon, the adventurer can no longer hover anywhere, and he must keep getting closer to the center of the black hole to feel the passage of time. Or simply, there is a wonderful corresponding relationship between the spatial dimension r in the distant viewer frame of reference and the time dimension τ in the adventurer frame of reference. For adventurers, r is no longer a spatial dimension that can move back and forth, but an one-way dimension like time.

Through the previous understanding of the Schwarzschild metric, we find that the event horizon of the black hole appears somewhere where the component of the metric is zero or divergent. According to this experience, it is natural to identify the event horizon of other types of black holes. The live-only, non-rotating RN metric goes like this:

Among them

Determining the position of the event horizon is very simple, it only needs to solve a small equation.

You get it.

These are the inner and outer horizon of the RN black hole.

When r-< r < r +, that is, the part between the two horizons, gtt > 0grad grr < 0, the spatial dimension r becomes an one-way dimension. In the region of r < r -, gtt < 0 and grr > 0, the time and space dimensions return to normal.

Readers who can persist in reading here are expected to have an illusion at this time-- what mysterious bending of time and space, that's all! Well, let's take a look at the Kerr metric without electricity and only with self-rotation angular momentum J.

Among them

Obviously, the space-time structure depicted by this metric is much more complex than the Schwarzschild metric and the RN metric, so it took longer to calculate. The Schwarzschild metric was discovered as early as 1915, and the RN metric was discovered in 1916-1918, but the exact solution of the Kerr metric was not until 1963.

The Kerr metric is not only complex but also important, because the celestial bodies in the universe have more or less rotational angular momentum, and only the Kerr metric can more accurately reflect the motion and evolution of these celestial bodies. In comparison, the Schwarzschild metric and the RN metric are oversimplified and even leave out a lot of interesting content in the real universe.

A rotating black hole can generate electricity in the Schwarzschild metric and the RN metric, and the position of gtt=0 happens to be the position of grr →∞, so this position is naturally defined as the event horizon of the black hole. However, in the Kerr rule, the positions that meet these two conditions no longer coincide.

By solving the gtt=0 equation, we can get

By solving this equation, we can get

These two sets of solutions correspond to four interfaces, which should be defined as the event horizon?

Recalling the previous description of the adventurer's journey, we can conclude two conclusions:

When the adventurer hovers over a position in space, the speed of his passage of time seems to be proportional to that of the distant viewer, and when gtt=0, the adventurer seen by the distant viewer comes to a complete standstill.

In the place where gtt < 0, the space dimension r becomes an one-way dimension like the time dimension t. The adventurer cannot hover at a point in space and must continue to move one-way along the r direction.

These two rules are universal to all time and space, and can also be used to examine the Kerr metric.

If the adventurer is allowed to go to a Kerr black hole, when he reaches the position of rE+, in the distance, he will look still, but in fact, he can move on, and even if he enters the area of r < rE+, there is still a theoretical possibility of escape. It is only when he enters the region of r < rH+ that he is really captured by the black hole and irrevocably dragged to the rH- position.

So the rH is the event horizon of the black hole, and the location of the rE is called the static surface, also known as the infinite redshift plane, and the light emitted from this position cannot carry energy into the distance. Careful readers may ask, how can adventurers escape from places where even the light cannot escape? This is related to the special geometric characteristics of this area between rE and rH.

In this region, gtt > 0 and grr > 0, it seems that both time and r dimensions have become "spatial dimensions". In order for adventurers to have real time τ, a negative contribution must be found in the line element expression.

It becomes the only source of contribution that can be counted on. It can be seen that in the region between rE and rH, Φ is a dimension that can not stop changing in one direction forever. This is the extreme drag effect of a rotating black hole on the surrounding space.

The space-time that rotates all the time is called the energy layer because it contains a special kind of energy. Adventurers entering this area can throw a mass object in the opposite direction of rotation, and they can get a lot of energy. and then use this energy to escape the static surface. The practice of getting power by losing a car is called the Penrose process process, which was discovered by Penrose in 1969.

We know that mass is a form of energy, and the Penrose process is essentially a way to convert mass into energy by using the energy layer of a Kerr black hole. And this kind of energy exchange is so efficient that up to 29% of the mass we feed into black holes can theoretically be converted into the energy we get. Although this efficiency is probably about the same as that of boiling water with coal, don't forget that our denominator is not the chemical energy in coal, but the mass of the whole mass.

The Kerr-Newman metric with angular momentum J and charge Q in the shape of the event horizon is a small extension of the Kerr metric.

It can be seen that there is not much difference between this metric and the space-time structure described by the Kerr metric, and we can also find the position of the static surface and event horizon through gtt=0 and these two equations.

And

Some additional instructions are needed here. The event horizon of both Kerr black hole and Kerr-Newman black hole is determined that rH is independent of coordinate parameters, which seems to be a sphere. But why is the event horizon painted in the shape of a pumpkin in the image given above?

In fact, this is also a result of the distortion of time and space. If you make rsqurh _ dtcounter _ zero, that is, to fix the time and position at the rH, the line element expression of the Kerr metric becomes

This expression obviously does not have spherical symmetry.

However, we cannot rudely say that this is the black hole "seen" in the eyes of adventurers, because it involves the route of light propagation, which will complicate the problem. We can only let the adventurer close his eyes, avoid the deception of the light, and touch the actual spatial position with his hands. Because of the severe distortion of space-time in the black hole, the adventurer will find that a spherical 3D indicator originally given to him by the distant viewer will turn into a flat pumpkin in the black hole.

There are always more questions than answers. There are too many research topics related to black holes. Except for the thermodynamics and information paradoxes of black holes, only many properties of space-time geometry itself are active research frontiers so far. For example, the singularity's destruction of the causal structure of time and space makes many researchers very uneasy: since it cannot be eliminated in theory, we very much hope that it will always be hidden in the event horizon and will not be exposed to the time and space that we can reach.

However, in the previous calculation of rH, we can see that if rQ or an is large enough, that is, the charge or angular momentum is large enough, then it is mathematically possible that rH is unsolvable, corresponding to the space-time structure in which there is no event horizon. If so, the singularity will be exposed in front of us, which is a scene that physicists resist very much. For this reason, Penrose put forward the "Universe Supervision hypothesis" (Cosmic censorship hypothesis), which holds that there must be some mechanism in the universe to prevent the emergence of naked singularities. As for what this mechanism is, there is no particularly powerful theoretical mechanism so far.

In addition, since it is known that a black hole is an ultra-high energy converter, the stability of its own structure has also become a concern for physicists. Just as the gunpowder workshop is more prone to explosion, a particle entering a black hole may gain a huge amount of energy from occasional decay, which may transform the particle itself into a small black hole. If such a process occurs, the small black hole may cause irreversible damage to the space-time structure of the large black hole, and even lead to the complete collapse of the overall structure of the large black hole.

The research on the structural stability of Kerr black hole is a very difficult subject, and the progress has been slow in the past 60 years since 1963. In May 2022, several researchers from Columbia University and Princeton University finally gave a mathematical answer in a 912-page paper.

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