A Man of Science and Engineering in Troubled Times Wandering in the Late Qing Dynasty

Chapter 608 Unbounded
After Li Yu received two letters from Einstein more than a month apart at the same time, he soon learned that Schwarzschild had obtained the first solution to the field equation of general relativity, a black hole.

As mentioned before, although the field equation looks harmless, it is actually a tensor equation with 10 unknowns. To be precise, it is a second-order nonlinear partial differential equation system composed of 10 equations!
Tensors are a very good mathematical tool, but unfortunately ordinary people want to understand it too much, and only a few pure science majors have access to it. Just know that it is a very difficult differential geometry equation to solve.

The name of differential geometry has been mentioned many times, and even Wei Shen is working on it. Two of the seven major mathematical problems of the millennium are also in the field of differential geometry.

There is no general solution to this equation, only special solutions.

That is to say, we need to set boundary problems, initial conditions, etc., and then obtain a corresponding special solution.

What Schwarzschild obtained was the first solution to the general relativity field equation in history, that is, the Schwarzschild solution.

It is also the most famous solution to the field equation, because Schwarzschild theoretically perfectly deduced the existence of black holes through general relativity.

Although black holes are still very mysterious in the 21st century, they are indeed nothing new.

As early as the 250th century, Laplace discovered through calculation that a bright celestial body with the same density as the Earth and a diameter times the sun will emit light that will be attracted to itself and cannot be seen by us.

So the brightest objects in the universe are probably invisible.

The last sentence is quite philosophical.

Laplace also gave the Schwarzschild radius formula of a black hole, which is r=2GM/cc (the last cc is the square of the speed of light).

The formula is correct, it is the same as that derived by later generations using the general theory of relativity, but Laplace's derivation process is wrong.

He calculated this by assuming light to be a particle. ——It is said that at that time, scientists on the European continent generally supported the wave theory of light, and Laplace was somewhat "deviant" in this point.

Schwarzschild's method is certainly correct at the moment.

At this time, Schwarzschild was serving in the German army, on the Eastern Front, confronting the Russian army.

Schwarzschild's speed was quite fast, and this result was obtained less than a month after Einstein published the field equations of general relativity.

When Einstein received this letter from the trench front, the old and wrinkled envelope was covered with dust, and the sender's name was covered by a large piece of blood. After opening it, he saw the name: Karl Schwarzschild.

"As you can see, war has been gentle to me. Although the fierce gunshots can still be heard not far away, please allow me to conduct this exploration in the garden of your mind."

——Unfortunately, the war was not gentle to Schwarzschild, and he died of illness a few months later.

During the calculation process, Schwarzschild simplified the initial conditions as much as possible, calculated the curvature of space-time outside a non-rotating spherical star, and then concluded that if all the mass of a star is compressed into a small enough space (later (called the Schwarzschild radius), then all calculations seem to be invalid, and space-time will bend itself infinitely.

For our Sun, this would happen if all its mass were compressed into a radius of less than three kilometers; for Earth, it would need to be compressed to about two centimeters, or about the size of a marble.

In this case, nothing within the Schwarzschild radius can escape the pull of gravity, not even light or other forms of radiation; time will be slowed to a standstill. In other words, to an outside observer, travelers near the Schwarzschild radius appear to be frozen in their tracks.

Because Schwarzschild died too early, there was no time to do more research.

Moreover, when the Schwarzschild solution was first proposed, it did not attract much attention. It was really difficult for people at the time to understand. How could there be a singularity with infinite density?
What the hell? !

Moreover, cosmology or astrophysics has not been developed to the corresponding level. At least we need to know about white dwarfs under electron degeneracy pressure and neutron stars under neutron degeneracy pressure before we can theoretically speculate on the existence of black holes.

This was a very long process, and it was only around 1939 that Oppenheimer sealed the deal (we still have to wait for astronomical observations in the future). After that, there will be further research on black holes by Hawking and others.

At present, even neutrons have not been discovered, so it is difficult to discuss black holes from the formation mechanism.

However, we can discuss some peculiar properties of black holes.

So Li Yu wrote an article about some interesting property predictions of black hole solutions under general relativity.

For example, many people know about the black hole event horizon: as long as matter enters the event horizon, it can only be sucked to the singularity.

Also, the space-time coordinates within the event horizon are interchangeable, and the event horizon is actually an isochronous plane. In a conventional sense, a circle, from the edge to the center, is a radius in space; but for a black hole, from the event horizon to the singularity, it is a time coordinate.

This property is quite interesting and important to consider carefully.

The singularity becomes the end of time, and time cannot go back, but can only move forward (the forward speed can change), so matter entering the event horizon can only run towards the singularity.

If you drive a spaceship and fall into a black hole, no matter which direction you increase the engine horsepower, it will only make you fall to the singularity faster, because that is the flow of time.

This leads to the fact that there must be a vacuum state between the event horizon and the black hole - everything falls into the singularity.

In addition, Li Yu also discussed gravitational redshift in the article.

He had mentioned redshift as early as his first visit to the Harvard Observatory.

To briefly review, redshift means moving away from us, which means that the wavelength becomes longer from a physical point of view.

It’s easy to understand. An example in life is a car driving towards us and then leaving. Driving towards us, the pitch will become higher, that is, the frequency will become larger, the wavelength will become smaller, and the blue shift will occur; moving away from us, just the opposite, the pitch will become smaller, that is, the frequency will become smaller, the wavelength will become larger, and the red shift will occur.

This is the most common Doppler red shift, but it is enough to know the principle and draw inferences.

Black holes cause gravitational redshift under the theory of relativity.

It’s not difficult to understand either.

We mainly discuss light.

First of all, remember that the photon's speed is set from the beginning of its birth to be eternal (without mass), and it will always be the speed of light c.

Black holes belong to a strong gravitational field, and photons have to pay a price to escape the constraints of gravity. Since the speed of the photon remains unchanged, according to the energy formula of the photon: E=hf (f is the frequency), it can only sacrifice a little of its own frequency.

——As the frequency decreases, the wavelength increases, which is a red shift.

Once you understand this, you can naturally understand gravitational time dilation.

Those who have watched the famous "Interstellar" should remember that in the process of searching for a planet suitable for human habitation in space, the protagonist once visited a planet under a strong gravitational field. He only stayed on the planet for a short time. Several days had passed outside. Year.

This is the slowing down of time in a strong gravitational field, or gravitational time dilation.

It is also not difficult to understand: the definition of time is the cesium frequency, and 1 second is 9192631770 cycles. Under a strong gravitational field, the frequency slows down and one second becomes longer.

The advantage of physics is that if you only discuss the physical meaning, it is not so profound, but it is also very interesting, it is easier to popularize science, and you gain a lot of insights.

Mathematics is relatively complex. It is purely a test of IQ. If you can do it, you can do it, and if you can't, you can't. The derivation process cannot be omitted or made wrong. If you don't pay attention to the analysis of the last big question in the college entrance examination, you will not be able to keep up, let alone more advanced mathematics...

Li Yu only discussed the generalization of these black hole properties, which is already very in-depth for the current field equations.

As for the subsequent rotating black hole (Kerr black hole), it is too complicated. This thing is a bit beyond cognition to modern people, because the center of the rotating black hole is not a singularity, but a strange disk.

Just the two words "Qipan" are difficult to understand.

What's even more outrageous is that if the black hole rotates fast enough, the odd disk of the Kerr black hole will become larger and larger, the event horizon may disappear, and the odd disk will be exposed from the event horizon.

Mathematically it is allowed.

But this is very scary: because the physical properties of the singularity are unknowable, if external observers can directly see the singularity, it will cause the causal relationship in space and time to be disordered.

Therefore, the famous general relativity expert Rogers later proposed a conjecture called the cosmic supervision conjecture, which means that there is a supervisor in the universe that does not allow such singularities to be exposed.

It sounds a bit like the god-level civilization in "The Three-Body Problem".

In addition to the content of black holes, Li Yu also wrote some conjectures about the universe under general relativity.

For example, the discussion of Einstein's sentence "The universe is limited but unbounded".

There are many things worth talking about in the theory of relativity, and this "universe is bounded but boundless" is a very key point.

Many people should have this confusion: How big is the universe? Is it infinite? If it is finite, what is outside the universe?
This question has troubled mankind since thousands of years ago, and now it has a preliminary answer.

"Limited" means that the universe is limited.

An absolutely infinite universe filled with stars and other objects would not be possible, because then every point would be attracted by infinite gravity, infinite light would shine in all directions, and the entire night sky would be bright.

This point has long been mentioned in the famous Olbers Paradox: If the universe is static and infinite, then the night sky should be as bright as the daytime, because the integral of the illumination of all stars must not converge.

As for "unbounded", that is, without boundaries, it can be understood this way: a finite universe floating at a random position in space is also unimaginable. If this is the case, how can stars and energy be maintained without dissipating from the universe and causing the universe to be exhausted?

Therefore, the universe can only be finite but without boundaries.

According to the general theory of relativity, the gravity of matter in the universe bends space. During the expansion of the universe, gravity causes space (actually the entire four-dimensional space-time) to completely fold back on itself, causing this system to be closed and limited, but without end or boundary.

That is, a three-dimensional hypersphere in a four-dimensional space.

Based on this conclusion, it is actually meaningless to discuss what is beyond this curved universe.

Just like a two-dimensional ant on a Möbius strip, it is impossible to understand what it is like outside.

Although we can use mathematics to guess the four-dimensional space, from the perspective of practical significance, it is really meaningless to ask what the world outside the three-dimensional space of our curved universe is like, let alone impossible to answer.

This article did not take much effort for Li Yu - any science and engineering student in the 21st century could complete it, and it could even be published in top journals such as "SCIENCE".

If you don't study it deeply, the theory of relativity will not be as tortuous as quantum mechanics.

Li Yu's writing is very popular, and many people will be able to understand it after it is published, and it is expected to cause a lot of discussion.

-

American Physical Society, President Michelson is organizing a seminar to discuss Li Yu's article.

"It's unbelievable. There is no one in the United States who understands the theory of relativity. Li Yu, as an Oriental, understands it so thoroughly." Michelson said in disbelief.

"And he was able to write such a high-quality article in such a short period of time," Millikan said. "Counting it in, we only saw the German Einstein article a week ago, right?"

Hale corrected: "According to the data, Einstein has become a Swiss citizen."

"Sorry, I don't pay much attention to theoretical physicists," Millikan said.

Michelson said: "Einstein is now in Berlin, working at the Prussian Academy of Sciences, and it took a lot of effort to get his papers out of Germany. We got the papers through Eddington of the Royal Society, not the official channel."

Millikan asked Hale: "I still have to ask again, is the theory of relativity really credible? I heard that the astronomy expedition team organized by Germany last year was captured in Crimea and did not complete the observations."

Haer, who is engaged in astronomy, nodded and said, "That's right."

Millikan said: "In other words, the theory of relativity is still just something that physicists have deduced on paper?"

"That's true," Hale admitted, "but there will be another solar eclipse in 1919. Then you only need to take a few photos, and it will be clear what is right and what is wrong."

Millikan obviously did not understand the theory of relativity very well: "How can the verification of physics be related to astronomy?"

Hale said: "Because the theory of relativity is inherently inseparable from astronomy."

Millikan glanced at Li Yu's paper, then looked at Einstein's, shook his head and said: "I spent several years doing experiments, and I just understood Einstein's photoelectric effect, but he doesn't do quantum theory. Did you learn the theory of relativity?”

Michelson smiled and said, "I don't need you to verify this time."

Millikan shrugged: "I don't understand astronomy either."

"Speaking of which, Li Yu from the East spans almost the entire range of physics, from classical thermodynamics to blackbody radiation, quantum theory, relativity, and astronomy," Michelson said, "and has made outstanding achievements in all of them."

Haier said: "His expertise in mathematics cannot be underestimated. Chaos theory and game theory are both major branches of mathematics with far-reaching connotations. In addition, he also has a large number of patents in engineering."

"Don't forget the popular Star Wars and Aliens," Millikan added with a smile, "I like watching them very much."

Michelson said: "Anyway, Li Yu is our member, and we can't find another person who understands the theory of relativity. I sent him a telegram and asked him to give a few lectures when he comes to the United States. We cannot lag too far behind Europe in cutting-edge theory." "You should at least understand what's going on, not to mention that Europe is still in a state of war."

(End of this chapter)