A few days ago, Wednesday 14th March 2018, one of the most legendary physicist of the modern times, Stephen Hawking, passed away aged 78. Tributes from all around the world flowed in, triumphing his great and creative mind that explored the edge of the universe and back despite physically confined to a wheelchair. The inspiring story of how he defied all odds survived his disease and went on to become one of the greatest physicist of all times is familiar to all and was even made into a movie. Personally, Stephen Hawking has been a great inspiration. I read the Brief History of Time when I was in high school, and was fascinated by the beauty of the universe and the elegance of the physics behind it. I was aspired to become a theoretical physicist. While at the end I chose the more practical route of electronics and semiconductor engineering, my interest in knowing about the universe have never faded and Stephen Hawking remains to be one of the people that I admire the most.

几天前，世界著名物理学家霍金与世长辞，享年七十六岁。这几天全世界都在悼念这个伟大科学家的身残志坚。虽然身体被禁锢在轮椅上，他的思想和灵魂却走到了时空的边缘，为世界解开宇宙的迷团。他在被医生判了死刑后却继续活出精彩人生的励志故事也被拍成了电影。对我来说，霍金算是我从小的偶像之一。我高中时看时间简史，就被那宇宙的奥妙和物理的优雅所着迷。虽然我最后选择了比较应用化的电子工程，我心里还是对基础物理有着一份热爱，而霍金始终是我仰慕的人。

Image source

So as my tribute to this great physicist of our times, I am going to write a series focused on his physics. This is the work that he dedicated his life to, and what he was passionate about. Only by understanding his work we could let our mind travel to the edge of the universe and back with him together, and let his legacy lives on.

所以为了纪念他，我决定要写一系列关于他的物理研究的文章，在自我学习之余希望可以为他科普一下他的毕生心血，让大家也可以在他的带领下走到时空边缘。

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Part 1: Black holes, thermodynamics and Hawking Radiation

第一章: 黑洞，热学定律和霍金辐射

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If we use two words to sum up Stephen Hawking’s research contribution, I believe it should be quantum gravity, not black holes. However, black hole was the tool that he used to show how quantum mechanics can be injected into the classical theory of Einstein’s General Relativity, to explain phenomena that seems to defy physics as we know it. So, let’s start our journey with his work on black holes.

霍金的研究基本上是围绕着量子引力。而黑洞只是他通往这个目标的途径。所以，在这第一章，让我们看看霍金对黑洞的研究。

Image source : NASA's Goddard Space Flight Center

One of his most famous contribution is the Hawking Radiation. To understand the Hawking Radiation, we first need to deep dive into a branch of physics that greatly our everyday life: The laws of thermodynamics. Thermodynamics is the study of the evolving energy of a system. There are four basic laws of thermodynamics that all physical system obeys. This can be summed up below.

霍金最有名的研究莫过于霍金辐射。要了解霍金辐射，我们从热学定律开始。一个物理系统里的能量变化取决于下图中的热学四大定律:

While the equations may look foreign, you probably have encountered a few of these laws in your high school science class. The first one, appropriately named the zeroth law of thermodynamics, simply states that the thermal energy of a system is absolute and can be measured by a reference unit, i.e. a temperature. The first law of thermodynamics is simply conservation of energy, expressed as the change of energy of a system must be equal to the change in thermal energy plus the work done on the system. Both of these laws are pretty easy to understand.

这些看上去很复杂的方程其实隐藏着很简单的原理。第一条，热学第零定律，其实就是说一个系统的热量是绝对的，所以可以用统一的方法，就是温度，去描述这个热量。而热学第一定律就是能量守恒定律，一个系统的能量变化一定是伴随着热量和功量的变化。这二条定律是中学程度的科学，我就不解释了。

The second law of thermodynamics considers what is called Entropy. Entropy by definition is the number of possible states that a system can have and can be loosely understood as a disorder of the system. The second law states that the entropy of a closed system would always increase. This roughly means that a system (including, of course, our universe), increases in disorder over time. An everyday example of the second law that I like to use is an air-conditioner. An air-conditioner cools the air in a room, which, in effect, decreases the entropy of the air molecules in the room. By second law, there must be an increase of entropy somewhere else, and this is proven by the fact that the exhaust from the air-conditioner is much hotter compared to the air around it. By including the exhaust in the close system, the second law of thermodynamics is obeyed. The third law of thermodynamic states that the entropy approaches a constant value as the temperature reaches absolute zero. That is really just saying that a system would approach its ground states – sometimes more than one – when the temperature approaches absolute zero. It also means that since entropy change is infinitesimally small approaching absolute zero, we can actually never get to absolute zero.

热学第二定律定义了一个叫熵的东西。熵其实是一个系统的形态的数量，越多形态，熵越高。也可以解释为一个系统的序列或混乱。越混乱，系统里有可能的形态越多，熵就越高。热学第二定律可以解释为一个系统的熵只会变高不会变低。例如一个开着冷气的房，房间变冷了，熵变低了。可是空调排的气绝对会比室外气温要高，因为室外的熵变高才能平衡室内变低了的熵。而热学第三定律叙述当系统接近绝对零度时，熵的变化会接近零。因为极低温时熵的变化很少，这意味着一个系统是永远不会到达绝对零度。

Image source: Science ABC

热学定律是物理基础定律，所有物理系统都应该遵守。可是黑洞却好像不然。一个掉进黑洞的物体，困在黑洞里的可能状态肯定比进黑洞前低，就是说整个宇宙的熵变低了。很明显这违反了热学第二定律，除非黑洞有可以变化的熵。可是黑洞是个引力场，没有温度啊！霍金和另外两个物理学家，卡特和巴丁，大胆的假设黑洞的熵和黑洞的面积是成正比的，而写下了黑洞热学四大定律。

Now as mentioned, these basic laws of thermodynamics should be applicable to all systems. However, this in the first glance does not hold true for black holes. Considering a particle that falls into a black hole. It is now confined inside the black hole. The number of configuration that particle can have is now less than that before it falls in. So the entropy of the whole universe has decreased. This clearly violates the 2nd law of thermodynamics. The only possibility is that black holes themselves have entropy that can change over time. This is the conclusion drawn by Stephen Hawking, Brandon Carter and James Bardeen, who co-developed a set of laws governing black holes known as the black hole. The integral part of the black hole laws of thermodynamics is that since black holes are simply part of the space time fabric, it does not have a temperature. Rather, the entropy is related to the surface gravity, or the gravity experienced at the event horizon (the point at which light cannot escape) of the black hole. The four laws of black hole mechanics, analogous to the laws of thermodynamics, is as follows:

The form of these equations is essentially the same as that of the thermodynamics. For example, the zeroth law simply states that the surface gravity is something absolute, similar to temperature, that can be used to quantify or describe a black hole. The first law states that a change in energy is a black hole must be accompanied by a change in the rotation, charge and area of the black hole, all of which provides “work done” that is required by conservation of energy. The second law, derived from the assumed relationship between entropy and the black hole’s surface area, (and the main contribution from Hawking), states that because of this, the area of a black hole must increase over time. A consequence of that is that when two black hole collides, they would merge instead of breaking apart. The third law states that the surface gravity of a black hole cannot go to zero, similar to the third law of thermodynamics which implied that absolute zero can never be reached.

这四大定律围绕着黑洞的引力而成，和热学四大定律对应。第零定律定义了黑洞的表面引力为绝对的，可以像温度一样去用来描述和比较不同的黑洞。第一定律是黑洞的能量守恒，黑洞的能量变化是它的表面积，自旋和电荷变化而引起。第二定律是说黑洞的表面积只会增长，用来对应热学第二定律的熵。而第三定律则叙述和温度一样的表面引力是不可能归零的。

Unfortunately, these laws are based on the assumption of that the area of the black hole is related to the entropy. Using classical treatment of black holes results in black holes having zero entropy, and the surface gravity of a black hole can be zero. At this point there is nothing linking these laws to reality. Hawking’s ingenuity was displayed here by recognising this discrepancy can be resolved by using quantum mechanics, in the form of what is now known as Hawking Radiation. Hawking Radiation relies on the uncertainty principle, ΔEΔt ≥ h/4π, which implies that a large energy or mass (mass and energy are equivalent by Einstein’s famous E = mc²) can appear over a very short time. This is the basis of what’s known as vacuum fluctuation where a pair of particle – antiparticle can just appear out of nothing, only to recombine and disappear into nothingness after a short time. Since in most cases these particle pairs only momentarily exist in a very short time scale, they are known as virtual particle pairs.

虽然这方程上的对应是十分优雅，可是在经典场论和广义相对论下，黑洞的温度是绝对零度，熵是零。在这前提下，黑洞面积和熵是不相通的。而霍金的高妙在于他看到的量子力学可以解决这个问题。在量子力学的框架下的海森堡的不确定性原理, ΔEΔt ≥ h/4π, 允许能量在真空中突然出现，然后在短时间后消失。而能量和物质可以在爱因斯坦的E=mc²下转换，所以不确定性原理允许一对正负粒子在真空中短时间的存在，称为虚粒子。

Image source: Neven Belic

现在设想一对正反粒子在黑洞的边缘（事件视界）出现。只中个一粒子有可能在视界内不能逃出而进入黑洞当中。因为能量守恒，另外一个粒子只能往黑洞逃出。对黑洞外的观察者来说，黑洞在发出粒子！这就是霍金辐射。而有辐射，就是有温度。用黑体辐射方程算出黑洞的温度是与黑洞的质量，也就是面积成正比。就是说黑洞的熵和面积成正比，和黑洞四大定律完全吻合！

However, if a virtual particle pair appear at the event horizon, there is a finite chance where one particle of the pair makes it out of the horizon while they other never escape. In order for energy to be conserved, this particle must be projected out of the black hole. This is the Hawking Radiation. Since a radiation implies a temperature via by the Blackbody radiation principle. And as the rate of Hawking Radiation is clearly proportional to the area of the event horizon, the area can therefore be related to the temperature, hence the entropy.

Stephen Hawking’s discovery of this equation closes the gap between the black hole thermodynamics and general thermodynamics, restoring the universal consistency of physics. So while a direct consequence of Hawking Radiation is that a black hole would eventually evaporate and disappear if it does not have other means in gaining its area (which means the second law of black holes need to be adjusted slightly), the true significance of Hawking Radiation is showing that quantum mechanics and General Relativity need to work together to explain phenomena that is otherwise unexplained. This may be the closest prove yet that a unified theory of everything does exist. In the next post, I will show you how Stephen Hawking used this knowledge on black holes and applied it to the physics behind the origin of the universe and the unified theory of everything.

霍金辐射最有名的结果，就是黑洞不黑，是会发亮的！而且黑洞会变小和蒸发（这让黑洞第二定律需要稍微的改动）可是霍金的真正突破在于展示了广义相对论和量子力学可以共存，而两者需要相助才能解释宇宙中的物理。这很可是统一万物理论确实存在的证明！下一章，我会带大家了解一下霍金怎样从黑洞的物理引申到宇宙起原和统一万物理论的发展。

References:

https://en.wikipedia.org/wiki/Black_hole_thermodynamics
https://en.wikipedia.org/wiki/Laws_of_thermodynamics
https://en.wikipedia.org/wiki/Stephen_Hawking
https://en.wikipedia.org/wiki/Quantum_fluctuation

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