By 苏剑林 | May 18, 2016
I am not a researcher of gravity, nor have I studied gravity in great depth. In terms of theoretical physics, I have studied classical mechanics and quantum mechanics far more than general relativity. Therefore, I probably shouldn't be talking about gravity to avoid misleading others. However, during a bus ride, the driver's braking and acceleration made me think of some things related to gravity that I found quite interesting. So, I’m sharing them here for everyone to enjoy and for experts to correct.
Gravity is more accurately called "Universal Gravitation." The term "Universal" (万有) has two meanings: 1. All objects can generate gravity; 2. All objects are affected by gravity.
Einstein found it extremely strange that a force could be "universal." This is the most distinct difference between gravity and the other three fundamental forces. In contrast, electromagnetic interaction only exists where there is "charge," weak interaction only exists between fermions, and so on.
Besides gravity, do we encounter any other "universal" forces in our daily lives? Seemingly not. But let’s imagine: when you are sitting on a long-distance bus traveling at a constant speed and the driver suddenly hits the brakes, at that moment, everyone leans forward. Not only that, your suitcase and personal belongings might move forward too. In fact, everything on the bus experiences a forward force! For the people and objects on that bus, at the moment of braking, there exists a "universal" force!
Let’s understand this accurately. From the perspective of Newtonian mechanics, there is actually no "real" force at all. Because we are on the bus, we naturally choose the bus as our reference frame. When braking, the bus becomes a non-inertial frame, and thus there is a corresponding non-inertial force (fictitious force). This force arises because we chose a non-inertial frame; it is not a "real" force (by "real," Newtonian mechanics means there must simultaneously be an agent exerting the force, an object being acted upon, and a method of action; for inertial forces, there is clearly no agent). From a mathematical perspective, this is an extra term generated by a coordinate transformation. Therefore, this force must be "universal." If we simply chose an inertial frame, this force would disappear.
However, while we are on the bus—and let’s assume further that we cannot see outside the windows—we might have no concept of an inertial frame, or we might believe that our current frame is an inertial one. Therefore, we are willing to believe that at the moment of braking, a "universal force" was indeed generated on the bus. Analyzing further, we find that this force originates from the braking, and the act of braking was performed by the driver. Thus, we might even conclude:
The driver exerts a universal force on everything in the bus!!
This conclusion sounds quite humorous. But if a group of people lived their entire lives on such a bus, they might well reach such a conclusion. Of course, with deeper research, the passengers would discover that it’s not that the driver directly exerts a universal force on them, but rather that the driver stepped on the brakes, which slowed down the whole bus, and the bus then pulled us along, creating the force. This is closer to the truth.
Now consider gravity and general relativity. Isn’t it exactly like this? Einstein felt that a "universal" force was inconceivable, so he proposed that it might not be a force at all, but likely just a consequence of a non-inertial frame. Thus, he first wrote down the Equivalence Principle:
Gravity and non-inertial frames are equivalent
(Textbooks might not phrase it exactly like this). He then went on to believe:
It is not that objects exert a universal force on other objects, but rather that objects act upon spacetime (just as the driver steps on the brake), and then spacetime acts upon other objects (just as the decelerating bus makes us lean forward).
This is the starting point of general relativity.
However, if it only stopped at this step, the theory of gravity would just be a re-statement of Newton’s Law of Universal Gravitation in a different way, without bringing anything new. Einstein, with a profound respect for beauty, required that physics satisfy the Principle of General Relativity.
The Principle of General Relativity holds that:
All reference frames are equivalent.
This is the most concise and economical requirement for physics!
Some readers might challenge this. If I sit in a car moving at a constant speed versus a car that is accelerating, the feeling is clearly different. How can one say they are equivalent? This is the difficult part of understanding general relativity. In fact, "reference frame" here refers to spacetime, not just space. We are accustomed to thinking about problems in three-dimensional space and separating time. But if you view space and time as a single entity—from the perspective of "spacetime"—they are equivalent. There is a similar example: when we say gravity causes spacetime to curve, some readers ask why they don't see their table bending. Similarly, it is "spacetime" curvature, not just "space" curvature.
As for how to derive the full theory of general relativity from the Principle of General Relativity, this article cannot cover that, but we can outline the logic. First, the Principle of General Relativity restricts the form of physical laws. Note that in general relativity, gravity is no longer a force; it is simply the structure of spacetime. Therefore, once the Principle of General Relativity is accepted, general relativity is actually the "simplest physical law"—it is equivalent to a free particle in Newtonian mechanics—because no other interaction forces have been considered yet (again, in general relativity, there are no other forces; so-called gravity is just the structure of spacetime. Wouldn't you say that not considering any forces is the simplest case?).
Because of this, by restricting the form of physical laws, one can derive general relativity without even knowing the specific content of those laws—because there really are no physical laws to speak of; in the Principle of General Relativity, we are merely describing the structure of spacetime before discussing forces.
$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi G}{c^4}T_{\mu\nu}$$As for the specific derivation? That requires the help of differential geometry. We can use a metric to describe spacetime, and the Principle of General Relativity essentially requires that physical laws possess general covariance. It is said that after Einstein had these ideas, he had to diligently study differential geometry before he could write down the complete theory. In fact, the entire derivation process also implies an unwritten convention:
When the new theory involves small masses and low speeds, it must reduce to classical mechanics.
After all, Newtonian mechanics has been well-tested by time.
Reviewing the entire article, it is really just a process of philosophical reflection. Yet Einstein, relying on such a process of reflection, arrived at general relativity (of course, the mathematical content of differential geometry cannot be ignored). This is exactly where Einstein's greatness lies!
Please include the address of this article when reprinting: https://kexue.fm/archives/3739
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,
author={Su Jianlin},
year={2016},
month={May},
url={\url{https://kexue.fm/archives/3739}},
}