This article was actually completed at the beginning of the year.
As is well known, we live in a flat world. As we can perceive, in this world known as "Euclidean flat space," the shortest curve between two points in space is a straight line segment. Any right-angled triangle in space satisfies the Pythagorean theorem. Every object has its own length, width, and height, and they all move as time passes. This worldview treats time as independent of space, as a unique object of study. However, since Einstein published Special Relativity in 1905, our universe has been described as a "four-dimensional spacetime" composed of three-dimensional space and one-dimensional time. Here, the status of time and space is equivalent. Many enthusiasts might feel very confused: Even if it is proven that there is indeed some connection between time and space, is it necessary to describe time as a dimension of the world? In our senses, time is clearly very different from the three dimensions of space. How can time and space be equated? Actually, the answer is simple: for the sake of beauty. By viewing time as a dimension equivalent to space, the entire system of mechanics exhibits an unprecedented symmetrical beauty. This beauty is not only pleasing to the eye but also greatly facilitates our further handling of problems.
Symmetry
Nature seems to favor symmetry, as seen in the symmetrical appearance of many butterflies, dragonflies, and birds. In China, we have had a special fondness for "symmetry" since ancient times, which can be seen in various symmetrical houses and pavilions in our lives. Not only that, it also appears in the works of literati. For example, the popular palindromic couplet "Guest at Heaven’s Abode, Abode’s Heaven at Guest" (Ke Shang Tian Ran Ju, Ju Ran Tian Shang Ke), and the poem "Autumn" from "Spring, Summer, Autumn, and Winter" by the Qing Dynasty poetess Wu Jiangxue:
Autumn river Chu geese rest on the sandbar,
Geese rest on the sandbar, shallow water flows.
Flowing water, shallow sandbar, sand-resting geese,
Sandbar-resting geese, Chu river in autumn.
It reads the same forwards and backwards; this is undoubtedly a wonderful palindromic poem!
What is symmetry? In our lives, there are many symmetrical phenomena, such as the shape of a basketball, the human body, and the elliptical orbits of planets around the sun. Symmetry is everywhere. The concept of symmetry is broad; the examples mentioned above are geometric symmetry, and there is corresponding algebraic symmetry. Taking the Pythagorean theorem of a right-angled triangle as an example, we know that the three sides of a right-angled triangle satisfy: a^2 + b^2 = c^2
Temporarily ignoring the geometric meaning of this formula and treating it only as an ordinary functional expression, let us swap the positions of a and b. What do we get? It is: b^2 + a^2 = c^2
Obviously, this is equivalent to the original expression. Thus, we say that in the expression a^2 + b^2, the "status" of a and b is equivalent; a^2 + b^2 is a symmetric expression. There are many other symmetric expressions, such as x+y, xyz, \sin(xy+yz+zx), etc. Swapping any two variables in these expressions does not change the expression. In a symmetric expression, every variable has equal status.
Now let’s look at the variable c in a^2 + b^2 = c^2. We find that after swapping a and c, the expression changes. Clearly, a and c here are not completely equivalent. But if we move c to the left side, we get: a^2 + b^2 - c^2 = 0
Using the imaginary unit i (i^2 = -1), the expression can be rewritten as: a^2 + b^2 + (ic)^2 = 0
In this way, in the expression a^2 + b^2 + (ic)^2, we have made a, b, and ic occupy completely equivalent positions. Since ic is just a linear transformation of c and does not change its nature, ic is essentially c. In other words, we have discovered a special equivalence relationship between c and a, b. In this sense, we should "view a, b, c with the same eyes."
Some enthusiasts might think that this is nothing more than a numerical game that neither brings computational convenience nor conforms to intuitive feelings, so why go to such trouble? But has the reader considered that once we establish the equal status of a, b, c, whenever we discover a law related to side a, we can effortlessly "convert" this law into a law for sides b and c (since their status is equal, natural laws will not favor one side over another), without having to recalculate for each side? This is undoubtedly time-saving and labor-saving. On the other hand, symmetry greatly helps us reveal the inner mysteries of natural laws. This is probably the benefit of symmetry.
Another example from physics is D’Alembert’s Principle. We all know Newton’s Second Law F=ma. D’Alembert rewrote it as F+(-ma)=0 and considered -ma to be a force as well. Consequently, all mechanical systems satisfy static equilibrium! A seemingly simple transformation actually represents a major step forward: using the principles of statics to study dynamics is undoubtedly more convenient and profound, because "static" is always easier to control than "dynamic." Of course, the author is only at a superficial stage of exploration and fears that his clumsy pen cannot fully describe its beauty, so the description in this area ends here.
Symmetry in Relativity
In 1905, known as the "Miracle Year," Einstein published the Special Theory of Relativity, which is still popular today. It derived the Lorentz transformation (taking one-dimensional space as an example): \begin{aligned}x' &= \gamma (x - vt) \\ t' &= \gamma (t - \frac{v}{c^2}x)\end{aligned}
I do not intend to detail the derivation of the Lorentz transformation here (interested enthusiasts can consult relevant relativity books; Wikipedia also has related content). Simply put, this is a coordinate transformation of an object between two different inertial frames. We all know that regardless of the chosen coordinate system, the length of a line segment does not change, which can be written mathematically as x^2 + y^2 + z^2 = x'^2 + y'^2 + z'^2.
Looking closely at the Lorentz transformation, we find the following fact: x^2 - c^2 t^2 = x'^2 - c^2 t'^2
This reminds us of the Pythagorean theorem. We rewrite it as: x^2 + (ict)^2 = x'^2 + (ict')^2
This is exactly the same form as x^2 + y^2 + z^2 = x'^2 + y'^2 + z'^2, and every quantity has the dimension of length! Thus, we find that time and space satisfy a certain invariance of "distance." Symmetry means conservation, and conservation means invariance. Here we have found an "invariant," which means we have found a kind of symmetry.
Now comes the main point. As mentioned before, x^2 + (ict)^2 and x'^2 + (ict')^2 are symmetric expressions. Here x and ict should have equal status. Therefore, if x is one dimension of spacetime, then ict should also be one dimension of spacetime. Since changing t to ict is just a linear transformation, the (mathematical) essence has not changed; therefore, time t also belongs to one dimension of spacetime. After Special Relativity was proposed, many people found it difficult to accept because it violated many classical physical intuitions. However, Minkowski’s reformulation made relativity no longer a "counter-intuitive" coordinate transformation, but a theory of spacetime with unified, symmetrical, and concise beauty. Many beautiful conclusions from the original three-dimensional space could be maintained or simply generalized. Moreover, because x^2 + (ict)^2 = x'^2 + (ict')^2 has the same form as the Pythagorean theorem, Special Relativity actually became a geometric theory! From then on, the world gained an extra dimension.
However, while space and time can form a four-dimensional spacetime, time and space are only equivalent mathematically, not entirely equivalent physically. For instance, time does not have the characteristic of being able to move freely like space. Therefore, the "spacetime" here is a mathematical "space" rather than a physical "space."
The Legendary Life of a Scientific Giant
The geometrization of Special Relativity mentioned above was the original creation of Einstein’s teacher, Hermann Minkowski (German: Hermann Minkowski, June 22, 1864 – January 12, 1909).
Minkowski said: "Einstein was a lazy dog during his student days. He didn’t worry about mathematics at all." Einstein’s Special Relativity indeed surprised Minkowski, but it was Minkowski who truly recognized the value of Special Relativity and advanced it significantly from a philosophical and mathematical perspective.
Hermann Minkowski was born in Russia, the youngest of three brothers. Interestingly, his family was undoubtedly extremely successful: his father was a successful Jewish merchant; his eldest brother inherited the family business and became an excellent businessman; his second brother was Oscar Minkowski, known as the "father of insulin"; later, his nephew Rudolf also became a famous American astronomer. Hermann Minkowski was known as a child prodigy in his youth due to his extremely high mathematical talent. At that time, the three Minkowski brothers were very prominent.
In 1873, Minkowski entered the Altstadt Gymnasium. It took him only five and a half years to complete eight years of schooling. After graduation, he first entered the local university, then transferred to the University of Berlin, and soon returned to the University of Königsberg. During university, he studied under mathematicians such as Helmholtz, Kronecker, Weierstrass, and Kirchhoff. At the University of Königsberg, Minkowski met his old friend Hilbert and they became close friends.
There is also an interesting story here. In 1884, the 25-year-old mathematician Hurwitz came to the University of Königsberg as an associate professor. Out of shared interests, he established a deep friendship with Minkowski and Hilbert. Every afternoon at five o’clock, the three of them walked in the apple orchard, discussing current mathematical problems—sometimes bowing their heads in deep thought, sometimes talking incessantly, sometimes arguing, and sometimes laughing heartily. To outsiders, they looked like a group of mathematical madmen. However, these discussions had a significant impact on the world of mathematics and physics. Hilbert later wrote: "In countless walks, the three of us explored every corner of mathematical science. Hurwitz was widely knowledgeable; he was always our guide."
During his university years, Minkowski won awards for his outstanding mathematical work. In 1895, he succeeded Hilbert as a professor at the University of Königsberg, and the following year he moved to the Swiss Federal Institute of Technology (ETH Zurich). During this period, the young Einstein was studying at the school and became Minkowski’s student. Two scientific geniuses met, and here they collided to produce brilliant scientific sparks!
In 1902, Minkowski was invited by Klein to transfer to the University of Göttingen as a professor of mathematics. After 1905, Minkowski devoted almost all his energy to electrodynamics. In 1907, Minkowski realized that non-Euclidean space could be used to describe the work of Lorentz and Einstein, combining time and space—previously considered independent—into a four-dimensional spacetime structure, namely Minkowski spacetime. Minkowski spacetime provided the framework for the establishment of General Relativity. In a 1908 lecture in Cologne, Germany, Minkowski proposed the concept of four-dimensional spacetime.
Based on this, different descriptions of the same phenomenon can be expressed in simple mathematical ways. This work provided the skeleton for Special Relativity. Nobel laureate Max Born once said that he found "the entire arsenal of relativity" in Minkowski’s mathematical work.
On January 10, 1909, Minkowski suddenly suffered from acute appendicitis. Rescue efforts failed (it is said due to anesthesia failure), and he passed away on January 12 at the age of only 45. That a genius of a generation should fall so innocently is undoubtedly a great loss to the scientific community. To commemorate him, the scientific community named asteroid No. 12493 "Minkowski."
The Secret Behind Geometrization: The Pursuit of Beauty
Why geometrize physical theories? Is physical theory inferior to mathematics? When Einstein first saw his teacher’s article, he was dismissive, thinking it would not bring any benefits. But later he discovered that only by highly geometrizing relativity could he construct General Relativity, which is universal to all reference frames and includes gravity. Therefore, he began to value his teacher’s results, gave them high praise, and consulted his classmates about Riemannian geometry and tensor analysis.
It turns out that to satisfy our pursuit of beauty, physical laws should conform to some symmetrical rules, and symmetry originates from geometry. Therefore, it is necessary to use geometry to describe physics. Furthermore, due to the maturity of geometric theory, it provides us with more powerful tools when dealing with physical problems or conceiving new theories. On the other hand, geometry is described by symmetry; a geometrized physical world implies a symmetrical, harmonious, and beautiful world, which makes us marvel at the wonders of the Creator! Also, this is actually a philosophical reflection: who knows if the physical world is not a geometric world? Perhaps the world was originally like this?
Unfortunately, Minkowski had already passed away by then and could not see his theory shine brilliantly under the creation of his prized student.
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