Fundamentals of General Relativity
General relativity, simply put, is two points.
- First, a matter with mass will bend the space-time around it. This is “matter tells spacetime how to bend”.
- Second, in the absence of external forces, an object always moves along a geodesic in spacetime. This is “spacetime tells matter how to move”.
There is no such thing as gravity here, there is no need for gravity at all.
This picture is like this. You can think of space-time as a spring bed. Originally, the spring bed was flat. Put a few balls on it, and the place where the balls are on the spring bed will become curved—these balls bend the space-time around them.
Why does the earth revolve around the sun? Newton thought that was because the sun had a gravitational pull on the earth. However, the general theory of relativity believes that the earth does not know where the sun is at all, but the sun bends space-time more severely, and the earth moves according to the geodesics of its own space-time. It’s like the small ball on the spring bed can roll around the big ball, and you know the big ball isn’t attracting the small ball, that’s just because of the depressions around the big ball on the spring bed.
In the same space and time, the speed of each object is different, and the geodesics they follow are also different. Some objects will fall directly to the sun, some will make elliptical motions around the sun, and some will pass by the sun. These are just objects moving along their own geodesics.
Of course, when each object with mass moves in the curved space-time, it is also bending the space-time around itself, but the degree of bending is different. The shape of space-time is determined by these substances, and all substances move along the geodesics of space-time around them.
Using a spring bed as an analogy is a last resort. The bending of space-time by matter is not the result of “pressing” down on the spring bed like a small ball, but the result of naturally bending the space-time in all directions around it. And please note that it is not only space that is bent, but also time, just this part, we will leave it to a later chapter to elaborate on it.
Here, I would like to clarify one more thing. You may have this question: since the mass of the high-speed moving object will increase, will the extra mass also bend space? The answer is no. The general theory of relativity that “matter bends space” can be understood as the “rest mass” of matter in the curved space, and rest mass is an invariant that all coordinate systems agree on. The inner geometry of spacetime is absolute, but spacetime is seen differently in different coordinate systems.
General relativity is that simple.
natural state of motion
Einstein once once more saw through the red dust. What is gravity? It can be said that there is no gravity at all, and there is only the curvature of space-time.
Or it can be said that the so-called gravity is the curvature of space-time that can only be seen on a large scale. The body of a whale is curvilinear, but when viewed up close, it resembles a flat facet everywhere on its body. A local geodesic is a straight, straight line, which is why we said in the last chapter that there is no local gravity.
At this point, we have to redefine the concept of “natural state of motion”. The so-called natural motion is the state in which an object is free and free without any external interference.
Aristotle believed that the natural state of motion is rest. This is in line with our lived experience—things that are not disturbed by external forces seem to stand still.
Later, Galileo and Newton said that this is not right, that force is not what makes an object move, it is actually what changes the state of motion of an object. An object slides on a smooth surface, and if no friction interferes, it will continue to do so. So there is no difference between constant velocity linear motion and rest, they are both natural motions.
And now, Einstein said, all motion along the geodesic is natural motion.
It can be imagined that there is a place in space that is very empty and without any stars. The space-time here is flat, and the geodesic line is a perfect straight line. Therefore, the movement of objects along the geodesic line is exactly the same as moving in a straight line with constant velocity.
If space-time is curved, the astronauts will revolve around the earth, and the out-of-control elevator will fall directly. These two motions may seem different, but they are both free fall motions, and they follow their own duties. geodesic motion. So although they have acceleration, they are still in natural motion.
There is no essential difference between free fall motion, constant velocity linear motion, and rest. No matter what experiments you do in a closed lab, there is no way to tell them apart. Einstein said that they are the same thing, they both move along the geodesic line, they are both natural motions.
Conversely, if you stand still on the ground and get tired following standing for a while, this is actually an unnatural movement. You wanted to fall down the geodesic line, but the floor stopped you. To experience true freedom, you should do free fall.
Why is the gravitational mass exactly equal to the inertial mass, and why do two iron balls, one light and one heavy, hit the ground at the same time? Now, general relativity provides an explanation for this coincidence—because geodesics are only related to the object’s initial velocity, not mass, as long as it’s not massive enough to be on par with the Earth to significantly affect the shape of space-time around it!
Look back at the two imaginary experiments described in the previous chapter. Whether you are on an accelerating rocket or standing still on the ground, there is an external force preventing you from following the geodesic, so they are the same.
Whether it’s a free fall near the Earth, or constant linear motion in an empty space without any stars, it’s a natural motion along that geodesic line, so they’re the same.
As long as you accept that the dimensions of space-time are relative, you can accept the special theory of relativity; as long as you accept that space-time can be curved, you can accept the general theory of relativity. After accepting these two properties of space-time, why does the speed of light remain the same, why is inertial mass equal to gravitational mass, and whether gravitational force actually exists…these questions no longer need to be entangled.
So, relativity is a simple theory, it’s just pretty deep; actually I think general relativity is easier to understand than special relativity, it’s just beautiful.
Maybe the next time you see a whale, you can think of general relativity.
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