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

Chapter 607 Field Equations

Although Li Yu already knew the results of Millikan's experiment, news from Germany was much slower.

Germany is a militaristic country, so it can fight very fiercely on the battlefield, and there are not many situations like Russia and France where the front line is tight and the rear is tight.

Berlin.

Einstein was working hard in his new apartment, and his hair increasingly resembled a signature chicken coop.

This apartment is in a good location in the city center, but it has no furniture. It only lives in Einstein, a bachelor - bachelor is a temporary term, because it is not far from the home of his future second wife, Elsa.

The apartment itself is spacious, with seven rooms, and is located on the third floor of a new five-story building. Einstein spent most of his time in his study, which contained a large wooden desk with piles of papers and journals piled messily.

For Einstein, 1915 was a troubled year, with war being the most obvious factor, and there was also considerable trouble in his own personal life.

The relationship between Einstein and his wife Mileva reached a freezing point, and it was difficult to even meet his children. He went to Switzerland and stayed for nearly a month, but only saw his sons twice. In his opinion, this is all "a mother's fear that her children will become too attached to me."

In fact, Einstein's two sons did admire him to the extreme. And his wife Mileva apparently couldn't bear to lose her child.

Under various distractions, Einstein used his ability to concentrate and was still able to devote himself to research and arrange all kinds of work in an orderly manner.

In fact, even in his research work, in 1915, Einstein was always filled with great anxiety and was sprinting towards the greatest achievement in his life - the general theory of relativity.

When Einstein first moved to Berlin, his colleagues at the Academy of Sciences thought he would set up an institute and recruit assistants to study the most pressing problem in physics—quantum theory. But Einstein has always been like a lone wolf in scientific research. He does not want to have a group of collaborators or students around him, just like Planck. Instead, he hopes to concentrate on what he cares about-the general theory of relativity.

It's just that after so many years, Einstein was obviously still stuck on a key point, and even suddenly discovered that there were major flaws in his theory. The flaw was fatal: Einstein discovered that the equation at hand was not generally covariant.

It can be said that all previous efforts are in vain. If it cannot be extended to non-inertial systems, all efforts will be in vain.

It was at this time that Einstein wrote a lot of letters and sent them to people such as Lorenz and Li Yu.

——It’s a pity that Li Yu received this letter very late.

At the same time, Hilbert of Göttingen was pressing forward. In Einstein's own words, you can almost hear his footsteps.

Although Hilbert's level of physics was far inferior to that of Einstein, his level of mathematics was far superior to that of Einstein, who almost failed in mathematics in college.

At present, the physical meaning of general relativity is basically clear, and the key lies in mathematics.

The two of them seemed to be running in a race.

However, Hilbert chose a relatively narrow track. Einstein chose a dual mathematics-physics approach, while Hilbert mainly adopted a mathematical approach that sought to find covariant equations.

Hilbert’s famous saying must be quoted again: “Physics is too difficult for physicists!”

Einstein encountered many difficulties in his research on general relativity, but one thing about him is very admirable: his perseverance based on his excellent physical intuition.

As everyone knows, quantum mechanics took a group of great minds over hundreds of years to slowly build; and the theory of relativity was almost completed by Einstein alone.

That’s why Einstein has such a high status in the world of physics. According to Landau’s ranking of physicists, there is only one T0 level, Newton;

There is only one T0.5, that is Einstein.

Bohr, Heisenberg, Dirac and other famous gods can only be ranked T1.

However, from another perspective, Einstein's research work was actually more lonely than that of most scientists.

Fortunately, no matter how doubtful others may be, he always believes in himself.

Einstein was not stubborn. By October (just over a month before he discovered the field equations of general relativity), he finally gave up his previous erroneous views and turned his attention to physical solutions (emphasizing his feeling for the basic principles of physics). ) shifted to rely relatively more on mathematical solutions, namely the use of Riemann and Ricci tensors.

When Li Yu came to Berlin to meet with Einstein before, he asked him this question.

But Einstein's research on Riemannian mathematics did not continue. At this time, he finally discovered that Riemannian mathematics can produce generalized covariant gravitational field equations.

Einstein's focus then shifted to the mathematical goal of finding generalized covariant equations.

How can I put it, mathematics has become Hilbert's best thing.

Near the end of 1915, Li Yucai received two letters from Einstein. In the first letter, Einstein said: "After my trust in the previous theory completely disappeared, I clearly saw that only through the general theory of covariance, That is, it is possible to find a satisfactory solution by combining it with Riemannian covariates. It’s a pity that I couldn’t stick to what you told me last year.”

After determining the direction, Einstein spent more than four weeks doing crazy deductions, playing with a lot of tensors and equations, constantly revising and renovating. And every week he gave lectures at the Prussian Academy of Sciences, during which he continued his calculations.

As of about November 11, his equations were still not generalized covariant.

- It will take another three weeks to get to this point.

On November 11, Einstein submitted his second paper.

He used the Ricci tensor in it, specifying new coordinate conditions that made the equations appear to be generalized covariant. But facts show that the problem has not been fundamentally solved.

Einstein was only one step away from the final answer, but he could never get past it.

This feeling is probably experienced by many people.

In desperation, Einstein could only send the paper to Hilbert again. Although he knew that the two were currently "competitors," Germany, which was blocked, could not contact anyone outside in the short term.

"If my current modification is reasonable, then gravity must play a fundamental role in the composition of matter," Einstein said helplessly, "but this will make my work more difficult!" - Gravity is One of the earliest forces discovered by human beings, it is also the most mysterious force. Before Li Yu's time travel, it was still full of unknown mysteries, such as the "graviton" that has fascinated countless physicists but has never been discovered.

If anyone can discover it, he will definitely win the Nobel Prize that year...

Hilbert's reply made Einstein very uneasy, because he said in the letter: "I am preparing to give an axiomatic solution to the major problem you raised."

It is very consistent with Hilbert's current axiomatic research direction.

However, Hilbert did not clearly state the progress of his research. He only said that he would discuss it when this physical research has made substantial progress.

(In fact, Hilbert got stuck too.)
But Hilbert still wanted to invite Einstein to come to Göttingen in person and explain his theory in detail in person.

The most interesting thing is that Hilbert added a provocative postscript at the end of the letter: "Based on my understanding of your new paper, the answer you gave is completely different from mine."

Hilbert's thoughts were certainly friendly, but Einstein did not have time to travel to Göttingen.

Sometimes inspiration comes so suddenly. A few days later, in a speech at the Prussian Academy of Sciences, Einstein suddenly realized that long-term quantitative changes lead to qualitative changes. At this moment, he finally overcame the most difficult step.

Eight years ago, when Einstein first began formulating his general theory of relativity, he stated a conclusion: gravity bends light.

This is one of the core ideas of general relativity. He calculated that the gravitational field near the sun would deflect light by approximately 0.83 arc seconds, which is consistent with the predictions of Newton's theory, which treats light as a particle.

He invited an astronomical expedition to Crimea to observe the solar eclipse to test his theory. However, due to the sudden outbreak of World War I, the expedition team was taken away by the Russian army while on the carriage and was unable to complete the observation. This move accidentally helped Einstein a big favor, because the data he gave at that time was wrong.

Using the revised new theory, which takes into account the effects of the curvature of space-time, Einstein calculated that light bends twice as much as previous calculations. Now, he predicts that the Sun's gravity will deflect light by about 1.7 arc seconds.

However, this prediction must wait until a suitable solar eclipse occurs again more than three years later before it can be tested.

But he was much more confident about this. After all, all the mathematical derivation was successfully completed. Unless there was no solar eclipse that day, the result would definitely be a victory for the general theory of relativity.

At this time, Einstein had not yet had time to compile the paper and publish it.

On the morning of November 11, Einstein received Hilbert's new paper, the paper he was invited to Göttingen to listen to.

Einstein was surprised and somewhat dismayed to see that the ideas in the paper were very similar to his own research.

So he replied to Hilbert with a concise letter aimed at affirming the priority of his own work.

Hilbert had no objections and generously said that he had no priority. "Sincere congratulations on your victory in perihelion motion," he wrote humorously in the letter. "If I can calculate as fast as you do, then in my formula, the electrons will have to be trapped, and the hydrogen atoms will give me Write an apology note explaining why it doesn't emit radiation."

But Hilbert is Hilbert after all, the leader in mathematics today. The next day, November 11, Hilbert sent a paper to a scientific magazine in Göttingen, announcing his own equation of general relativity.

The title he chose for this paper was anything but modest: "Fundamentals of Physics."

These great scientists are quite interesting. A mathematician is writing "Basics of Physics".

After a period of compilation, on November 1915, 11, a date worthy of being engraved on the monument of physics, Einstein proposed a set of covariant equations in his last lecture on "Field Equations of Gravity", which made the general theory of relativity reach peak.

The beautiful equations of general relativity finally appeared in front of the world.

To ordinary people, this result is not as vivid as the mass-energy formula. But using simple mathematical tensor notation, all kinds of complexity can be incorporated into subscripts, and the final field equations of Einstein's general relativity are very compact and concise.

This is a truly generalized covariant equation that encompasses all forms of motion, whether inertial, accelerated, rotational or arbitrary.

Regarding Hilbert and Einstein’s “race to general relativity”, I have mentioned part of it before, and it is worth saying a few more words here.

Although Hilbert's equations are similar to the final version of Einstein's lecture on November 11, there is one key difference:
Hilbert's equations are not truly generalized covariant, and Hilbert did not shrink the Ricci tensor and put the resulting Ricci scalar into the equation. But Einstein did so in his speech on November 11.

In addition, later research found that Hilbert only submitted the final version of the paper in December.

Apparently Hilbert made corrections in the revised version of the article to conform to Einstein's version. And speaking of gravity, he very magnanimously added the phrase "first introduced by Einstein."

There is a brief debate among experts on the history of physics over the credit given to Hilbert and Einstein, but they all agree that the physical theory behind the field equations of general relativity should be attributed to Einstein.

"Hilbert discovered the last few mathematical steps almost simultaneously and independently with Einstein, but almost everything before them is due to Einstein," said one expert on this history of physics. "Without Einstein, it might have been decades before the laws of gravity of general relativity were discovered."

The broad-minded Hilbert also thought so. He clearly stated in the final published version of the paper: "In my opinion, the resulting differential equation of gravity is consistent with the magnificent general theory of relativity established by Einstein."

He has since acknowledged that Einstein was the sole creator of the theory of relativity.

But - there is still a but, Hilbert said: "Every kid on the street in Göttingen knows more about four-dimensional geometry than Einstein. But despite this, it was Einstein who did this work Stan, not the mathematicians.”

In terms of mathematics, Hilbert was still quite dissatisfied with Einstein.

——In fact, if Einstein hadn’t told the big guys in Göttingen a lot of his research reports, Hilbert might not have had the intention to study the field equations of general relativity.

Of course, Einstein and Hilbert himself had always had a pretty good relationship.

It happened to be World War I, and the two of them were the only two of the leading figures in the German scientific community who opposed the war. It is not an exaggeration to say that they sympathized with each other.

In the end, it was too late for Li Yu to receive the letter.

He received two letters from Einstein at the same time. The first was a letter complaining about mathematical difficulties; the second was a letter happily telling Li Yu that he had finally completed all the work on the general theory of relativity.

Li Yu was extremely moved, the other person was Einstein!
After seeing the field equations, Li Yu was finally able to write something new.