The sensation of gravity on Earth is merely the feeling of acceleration from the ground's reaction force.

When I learned about general relativity in a popular science magazine as a child, it came as a huge shock. Of course, I couldn't follow the mathematical formulas carefully back then, but Einstein's idea that spacetime is curved by the mass of objects, and that this creates gravity, seemed very fascinating even to my young self.

Einstein completed the general theory of relativity using only a notebook and pencil, relying on a few natural assumptions without performing any experiments or observations himself. The most important of these assumptions was the 'equivalence principle'.

We who stand on Earth feel gravity, but people in free fall high above do not feel that gravity. Rather, it feels like floating weightlessly in space. Einstein grasped this deeply and considered that, in fact, for an observer in free fall, Earth's gravity does not exist. In other words, he posited that gravity can be eliminated by the observer's motion. Furthermore, future large space stations will generate artificial gravity inside through their rotational motion. From Einstein's perspective, this artificial gravity is also equivalent to real gravity, with no distinction between them. This is the equivalence principle. Using this principle, Einstein completed the general theory of relativity as a theory of gravity.

However, when I learned about this, something remained unresolved in my young mind. The artificial gravity of a space station is easy to understand, but in the case of actual Earth gravity, there is the ground, and I thought that we, who are stationary on it, do not have the sensation of undergoing accelerated motion. On the round Earth, the distance between a person in Japan and a person in Brazil, on the exact opposite side, is constant. If they were undergoing normal acceleration, the distance between those two should be increasing rapidly. However, in reality, that does not happen. Something remained that I couldn't quite accept as a young person back then.

However, now that I have properly studied general relativity, I understand that we are indeed continuously accelerating at every moment. The reason it didn't fit my young intuition was that I was still clinging to the image of flat spacetime. For example, let's prepare two electrons in flat spacetime. Then, due to the electrical Coulomb force, the two will repel each other. As a result, they gain acceleration as shown in Figure 1, their velocity changes, and the distance between the two electrons increases over time.

However, what general relativity teaches is the fact that the massive Earth curves spacetime. As mentioned in the article below, when spacetime itself curves, the trajectory of particles moving straight also appears curved as a whole.

The same applies to electrons on Earth. If an electron had no electric charge, it would continue to move straight toward the center of the Earth. In Newtonian mechanics, this is perceived as a fall caused by the action of gravity, but in general relativity, it is different. From the equivalence principle, gravity does not exist for this electron, and the electron is moving without any force acting upon it. Accelerated motion is not occurring in the coordinate system fixed to the electron.

In reality, an electron has an electric charge, and an electrical repulsive force—that is, a reaction force—acts upon it from the electrons and atomic nuclei that make up the Earth. And it is possible to stop an electron on the surface of the Earth. But in fact, that situation is one where the reaction force from the ground is accelerating the electron. As shown in Figure 2, we are simply mistaking this acceleration caused by the reaction force for something caused by 'gravity'.

Even with the reaction force of the ground, the reason the distance between electrons placed in Japan and Brazil—which should be accelerating—does not change is because that spacetime is curved. The two electrons are being pushed from the ground and are constantly accelerating, but because spacetime is curved by the Earth, there was no change in the distance between the two electrons as a whole. Being able to understand such things accurately is one of the joys of studying physics.

However, it is an important perspective that the electric force here can also be derived from the general relativity of higher-dimensional gravity.

The gauge potential of the electromagnetic field can be described as part of the metric tensor of a higher-dimensional gravitational field. Then, the story of the antagonism between a particle's gravity and electric force in a black hole as a 4D spacetime, as mentioned above, can be written in a unified way in the general relativity of higher-dimensional spacetime, and just as gravity was, the electromagnetic force also originates from the geometry of spacetime. Such a perspective is also very important for us who stand on the ground and feel gravity.