數學與宇宙的神奇超乎想像。
為了理解宇宙，科學家常需要考慮那些極端的異常情況。南安普敦大學數學物理學家 Carsten Gundlach 說：「我們總是需要考慮極端情況，就是那些位於邊緣的特例。」黑洞就是宇宙中神秘莫測的極端。根據愛因斯坦的廣義相對論，黑洞之中的物質非常緻密，任何東西都無法逃離。幾十年來，物理學家和數學家一直在藉助黑洞來探究他們有關重力和時空的想法的極限。但即使黑洞，也有邊緣特例 —— 而這些特例也能為我們帶來不同的洞見。黑洞會在空間中旋轉。當物質落入其中時，黑洞的旋轉速度會加快；如果該物質帶電，則黑洞也會帶電。原理上講，黑洞所帶的電荷量或自轉速度都有一個極限，這應該取決於其質量。這樣的黑洞被稱為極端黑洞（extremal black hole）－ 它們是極端的極端。這些黑洞具有一些古怪的性質。尤其值得一提的是：這些黑洞的邊界（即事件視界）處的表面引力為零。 Gundlach 說：「這是一種表面不再吸引任何物質的黑洞。」但如果你把一個粒子輕輕推向該黑洞的中心，它也無法逃離。 1. 1973 年時，著名物理學家史蒂芬・霍金、約翰・巴丁（John Bardeen）和布蘭登・卡特（Brandon Carter）斷言真實世界中並不存在這樣的極端黑洞 —— 它們根本不可能形成。儘管如此，過去五十年來，極端黑洞依然是理論物理學領域的一個有用模型。羅德島大學的 Gaurav Khanna 說：「它們有非常好的對稱性，能讓運算更簡單。」這讓物理學家可以檢驗那些關於量子力學和重力之間神秘關係的理論。
史蒂芬‧霍金
兩位數學家證明霍金等人的論點是錯誤的。這兩位數學家分別是麻省理工學院的 Christoph Kehle 和史丹佛大學的 Ryan Unger。他們最近透過兩篇論文證明我們已知的物理定律並不能阻止極端黑洞的形成。
論文1：Gravitational collapse to extremal black holes and the third law of black hole thermodynamics; arXiv:2211.15742論文2：Extremal black hole formation black hole 15742論文2：Extremal black hole formation. Dafermos（也是Kehle 和Unger 的博導）說，他們的數學證明「很美，有著技術上的創新，並且有著出人意料的物理學結果」。他補充說，這暗示著宇宙可能比之前人們認為的更加豐富多樣，「在天文物理學上，極端黑洞可能存在」。但這並不意味著它們真的存在。 「就算存在一個性質優良的數學解，也不一定意味著大自然就會用到它。」Khanna 說，「但就算我們以某種方式找到了一個，也會讓我們思考我們忽略了什麼東西。不可能定律在 Kehle 和 Unger 的證明出現之前，我們有充分的理由相信極端黑洞不可能存在。 1973 年，巴丁、卡特和霍金提出了四個有關黑洞行為的定律。它們類似於四個長期確立的熱力學定律 —— 一組神聖的原則，例如：宇宙隨著時間的推移變得更加無序、能量既不能被創造也不能被毀滅。
Mathematician Christoph Kehle, who recently overturned the 1973 conjecture about extreme black holes. In a paper, these three physicists proved the first three laws of black hole thermodynamics: the zeroth law, the first law, and the second law. By extension, they assumed that the third law (similar to its counterpart, the standard law of thermodynamics) was also true, although they could not yet prove this. This law states that the surface gravity of a black hole cannot drop to zero in a finite amount of time—that is, it is impossible to create an extreme black hole. In support of this assertion, the three physicists said that if a process could bring a black hole's charge or spin speed to a limit, then that process could cause the black hole's event horizon to disappear completely. It is generally believed that there is no black hole without an event horizon, that is, a naked singularity. In addition, because it is known that the temperature of a black hole is proportional to its surface gravity, a black hole without surface gravity would have no temperature. Such a black hole would have no thermal radiation—and Hawking later proposed that black holes must emit thermal radiation. In 1986, physicist Werner Israel published a proof of the third law that seemed to put the question to rest. Suppose you want to create an extreme black hole based on a regular black hole. You can make it spin faster, or you can add more charged particles to it. Israel's proof seemed to show that doing so would not force the black hole's surface gravity to drop to zero for a finite time. As Kehle and Unger eventually discovered, Israel's argument concealed a flaw. The Death of the Third Law Kehle and Unger did not originally set out to find extreme black holes. Their discovery was entirely accidental. They were studying the formation of electrically charged black holes. "We realized we could create black holes with all charge-to-mass ratios," Kehle said. This includes the case where the charge is as high as possible, which is the case of an extreme black hole.
After proving that highly charged extreme black holes are mathematically possible, Stanford's Ryan Unger has now set out to try to show that the same is true for rapidly spinning black holes. But the problem is much more difficult.
Dafermos recognized that his former students had discovered a counterexample to Bardeen, Carter, and Hawking's third law: their research showed that a regular black hole could be turned into an extreme black hole in a finite amount of time.
Kehle and Unger's proof starts with a non-rotating, uncharged black hole and then models what happens when it is put into a simplified environment called a scalar field. Scalar fields assume the presence of uniformly charged particles in the background. They then blasted the black hole with pulses from that field, adding an electric charge to it.
These pulses also provide electromagnetic energy to the black hole, thereby increasing its mass. The two mathematicians realized that by sending diffuse, low-frequency pulses, the black hole's charge would increase faster than the black hole's mass—which was exactly what they needed to complete their proof.
After discussing the results with Dafermos, they pored over Israel's 1986 paper and discovered the error. They also constructed two other solutions to Einstein's equations of general relativity, which involve different ways of adding charge to a black hole. They verified the conjectures of Bardeen, Carter, and Hawking in three different situations, and obtained conclusive results. "The third law is dead," Unger said.
The two also proved that the formation of extreme black holes does not lead to a naked singularity, as many physicists feared. Instead, extreme black holes appear to be at a critical threshold: Add just the right amount of charge to a dense cloud of charged matter, and it collapses to form an extreme black hole. If this amount is exceeded, the matter cloud will not collapse into a naked singularity, but will spread out. No black hole will form at all. This result is exciting to Kehle and Unger because it proves that extreme black holes may exist.
Columbia University mathematician Elena Giorgi said: "This is an excellent example of mathematics giving back to physics."
Once impossible, it is possible today
Kehle and Unger It proves that extreme black holes can theoretically exist in nature, but it does not guarantee that they will exist.
First, those theoretical examples have large amounts of charge. But humans have never observed an obviously charged black hole. The likelihood of finding a rapidly spinning black hole is much greater. Beyond the charge version example, Kehle and Unger wanted to build an example where rotation reaches a threshold.
But the mathematical difficulty of studying rotation is not the same. “In order to do this, you need a lot of new math and new ideas,” says Unger, who and Kehle are just starting to work on the problem.
At the same time, if we can better understand extreme black holes, it will also help us better understand near-extreme black holes - which are believed to exist in large numbers in the universe. "Einstein once thought black holes couldn't exist because they were so weird," Khanna said. "But now we know there are black holes everywhere in the universe."
For similar reasons, he added : "We shouldn't give up on extreme black holes. I just think there are no limits to nature's creativity."
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