A Dialogue with Sir Roger Penrose
It was a sunny day; I had just finished my affairs in Jericho and was heading to the Museum of Oxford to attend another event. I was walking along Woodstock Road as usual. Yet, an old man sitting peacefully on a bus bench drew my attention. I pondered to myself, “Is this Sir Roger Penrose?” I had been reading his book a few weeks ago, and his interpretation of black holes and time was mind-blowing and amazing. I couldn’t believe my eyes that I was encountering such a huge figure in this casual setting. I pinched my skin to make sure I wasn’t in a dream.
I gathered my courage and crossed the road. Though I was very nervous, I managed to approach him. At first, I stood behind the billboard at the bus stop, peering into his face to double-check if he was Sir Roger Penrose. The more I looked at his face, the more determined I became. Finally, I was spunky enough to say to him, “Excuse me, are you Sir Roger Penrose?” He answered, “Sorry?” I thought he might have a hearing difficulty considering his age; Sir Roger Penrose was 93! I tried again, this time with a higher volume, “Are you Roger Penrose, Sir?” He looked up at me and answered with a gentle voice, “I am afraid so.”
Suddenly, my heart was pumping fast and beeping; I realized that the man in front of me was Sir Roger Penrose, widely regarded as one of the greatest thinkers and scientists in the world. He had made significant contributions to understanding the nature of the universe. Despite being an intellectual giant, he dressed and acted like nobody, and friendly smiled at me, leaning on his walker without any companions. One might simply think he was an ordinary old man, going out to shop for his dinner.
“I started by asking some questions about black holes, specifically about whether there is entropy in black holes. He responded affirmatively, explaining that he had collaborated with another renowned British scientist, Stephen Hawking. Together, they worked on the thermodynamics of black holes, considering the entropy associated with them. They proposed that the total entropy of the universe could never decrease, even when black holes form and swallow matter.
Entropy and Black Holes
In classical thermodynamics, entropy (S) is a measure of the disorder or randomness of a system. The definition of entropy can be approached in two ways. In classical thermodynamics, entropy is calculated in terms of macroscopically measurable physical properties, such as bulk mass, volume, pressure, and temperature. However, statistically, entropy is defined in terms of the statistics of the motion of the microscopic constituents of a system, modelled classically by Newtonian particles and later by quantum mechanics.
The formula can be summarized as:
H(X) = — Σ P(x) * log2(P(x)).
This information formula provided a quantitative measure of information, enables us to quantify the amount of uncertainty or surprise associated with the random variable or a set of data. Let’s consider a random variable X representing the colour of an object in a cubicle, and the there are three possible colours: red, blue and green. The probabilities of each color are given:
P(red)=0.4
P(blue)=0.3
P(green)=0.3
Now applying Shannon’s information entropy formula to calculate the entropy of this random variable: we will have H(X) ≈0.5838.
Therefore, the entropy is relatively low, given that the colour of an object is somewhat predictable. Also, the significance weights indicate that one needs approximately 0.5838 bits on average to convey the colour information and the probability of the random variable. In this case, if you were to guess the colour of an object in the cubicle based on the provided probabilities, you would have a better chance of being correct.
Yet, Shannon’s entropy is a measure of the uncertainty or information content associated with a probability distribution or a random variable. It is closely linked with communication theory and information science. In the context of a black hole, the concept of entropy is often associated with gravitational interaction, quantum mechanics, and thermodynamics; thereby, it is related to the statistical mechanics of microstates near the event horizon.
Therefore, we need to consider the formula at the microstate level. It can be written as follows:
S= k. log W
Let S represent the entropy of the system, k denotes the Boltzmann constant, and W signify the number of microstates corresponding to the macroscopic state of the system.
Boltzmann’s argument, rooted in statistical mechanics, supports the idea that entropy tends to increase over time, consistent with the second law of thermodynamics. This law states that in a closed system, entropy either increases or remains constant, but never decreases.
Building upon these principles of statistical mechanics, Stephen Hawking, in collaboration with Roger Penrose, made ground-breaking contributions to the field of black hole physics. They extended the application of statistical mechanics to black holes, proposing that these cosmic entities also exhibit entropy based on their microstates.
Subsequently, let’s discuss the formation of black hole then, which could be summarised according to the Tolman-Oppenheimer-Volkoff(TOV) equation:
dP/dR=-((ρ+P/c² )(m+(4πr³ P)/c² ))/(r(r-2Gm/c² ))
Where P is the pressure inside the star, ρ is the energy density inside the star, m is the mass enclosed within a sphere of radius r, G is the gravitational constant and c is the speed of light. The term (r-2Gm/c² ) is crucial in this equation, of which 2Gm/c² is Schwarzschild radius, which is the radius of non-rotating black hole. As the radius (r) approaches this critical value, the density increases indefinitely, causing gravitational forces to collapse the matter into a singularity — a point that draws everything inward. Due to the intense gravitational pull, even light is unable to escape. In this sense, this object has been named as “ black hole”, indexing that it does not emit any detectable radiation in the form of visible light or other electromagnetic waves.
However, American physicist Jacob Bekenstein proposed that black holes possess entropy, suggesting the potential existence of radiation. He posited that the entropy (S) of a black hole is directly proportional to the area of its event horizon, as described by the formula:
S=(kAc³)/4Gℏ
Renowned British scientist Stephen Hawking discovered that black holes emit thermal radiation as a result of quantum effects near their event horizon. The Bekenstein formula indicates that as the event horizon size increases, so does the entropy. This concept has established the premise that black holes possess a certain temperature. Notably, the temperature (T) of this radiation is inversely proportional to the mass of the black hole.
T=(ℏc³)/8πGMk
This notion emerged as a direct consequence of this statistical mechanical viewpoint applied to black holes. The radiation is a manifestation of virtual particle-antiparticle pairs being created near the event horizon, with one particle falling into the black hole and the other escaping as radiation. This process reflects the intricate interplay between quantum mechanics, general relativity, and statistical mechanics, providing a profound link between the microscopic world of particle interactions and the macroscopic behaviour of black holes.”
Time
After this conversation, I eagerly delved into a discussion about the nature of time. In presenting my perspective, I drew upon my philosophical reflections, specifically influenced by German Philosopher Martin Heidegger. I referenced the concept of temporality, positing it as the fundamental mode in which human existence (Dasein) unfolds. According to this viewpoint, we experience time as a horizon or framework within which all our activities occur. In essence, Heidegger’s philosophy on time intricately intertwines with the very essence of human existence.
Das Verstehen des Seins ist selbst eine bestimmte Weise des Seins, und dieses Sein des Daseins konstituiert sich in seiner Zeitlichkeit. Darin erhält es seinen ontologischen Sinn. Unsere Untersuchung wird von der Frage nach dem Sinn des Seins geleitet und zielt darauf, die Strukturen des Seins, die zur Konstitution des Daseins gehören, transparent zu machen.
( The understanding of being is itself a definite way of being, and this being of Dasein is constituted by its temporality. In this it gets its ontological meaning. Our investigation is guided by the question of the meaning of being and aims at making the structures of being, which belong to Dasein’s constitution, transparent.)
However, I contend that one ought to incorporate consciousness into the understanding of human expressions. While the concept of human pertains to an object, consciousness is the force shaping our reality. Therefore, a dynamic interplay should be established between time and the existence of human consciousness.
Indeed, all human comprehension of the universe hinges on the presence of human consciousness and its intricate interaction with nature. Consequently, the notion of an absolute and inherent reality is elusive, as crucial measurements and quantifications are constructed upon our sensory perceptions. Consider the vastness of the universe versus the minuteness of an ant — these size perceptions are entirely contingent on our subjective interpretation. Given the constrained capacity of human intelligence and sensory faculties, we may be confronted with an overwhelmingly “complex or simple” universe, obscured by these inherent limitations and the absolute objectivity does not exist to human.
At this elevation, I shared my perspective on the nature of time alongside others:
I posit that time serves as an interpreter of events in the universe, which comprises an infinite array of occurrences, ranging from the movement of atoms to the eventual demise of the entire cosmos. From a human perspective, time imparts reason and meaning to all events.
The inherent nature of time suggests that its rate should be infinite. Let’s consider the duration of time moving from point A to point B as Δt, and denote the change in its status as Δι. Given that Δt is infinitesimally small, and the purported change in status (distance between point A and point B) is a constant ‘a’, in this context, the rate of time becomes infinitesimally large, which can be summarised:
lim┬(a→0)〖(d(∆t))/a〗=⋈
However, there are those who contend that time is not a tangible entity but merely an illusion created by our perception. Sir Penrose found humour in this idea, asserting that time must indeed exist for us to make sense of things. Nevertheless, he acknowledged that the nature of time is a intricate blend of physics, psychology, reality, and philosophy. The perception of time varies depending on the observer, which explains why one may not feel its passage or sense it is accelerating while asleep.”
In Special relativity, where an observer a is moving at a constant velocity to another observer, the Lorentz transformation equations can define the changes of time between two inertial observers. The stationary observer experiences the measured time as t, while the moving observers as t’, t^’=t/γ, in which γ=1/√(1-v³/c³ ).
After this intriguing conversation, Sir Roger Penrose headed to his appointment, while I made my way to mine. He asked if I was Chinese, and I responded affirmatively. This led to a discussion about Yang Chen-Ning, a renowned Chinese theoretical physicist, who, in 1957 at the age of 35, was awarded the Nobel Prize in Physics alongside his collaborator Tsung-Dao Lee for their ground-breaking work on the violation of the principle of parity in weak nuclear interactions. Until then, it was believed that the laws of physics were symmetrical with respect to mirror reflection, a property known as parity. In 1956, Yang and Lee demonstrated that the weak force, responsible for certain types of radioactive decay, did not follow this law. In China, this is famously referred to as 宇称不守恒 (the violation of parity).
Their theory was later experimentally confirmed by Chien-Shiung Wu, highly regarded as the queen of physics and known as “Chinese Miss Curie.” This achievement was applauded by many Chinese as it boosted morale during a time when Chinese society, technology, and the economy lagged behind. Yang is praised as a Chinese hero. Sir Roger Penrose calmly asked me if Yang was over 100, and I confirmed. I shared with him Yang’s influence and achievements in science, to which he responded, “Of course, I have met him before.”
As we parted ways in a small corner, this brief conversation left a profound impact on me. Sir Penrose, one of the greatest mathematicians and physicists, a living scientific legend who witnessed significant events post-World War II, displayed humility and amiability, akin to conversing with a grandfather. Remarkably, his mind functions incredibly sharply at his age, The most profound questions find the most succinct and effective answers from him. His enthusiasm and passion for science and truth touched my heart deeply. I believe his success stems from unwavering passion for his career, transcending worldly concerns such as social titles, honours, money, and power.
I stood behind him, watching as he walked away ploddingly. In that moment, a multitude of thoughts flooded my mind. Among them, I seized upon a particular idea — an interpretation of success attributed to Albert Einstein: A=X+Y+Z, where X represents work, Y is play, and Z is the art of keeping one’s mouth shut!
