_{1}

The uniformly accelerated motion is studied in the framework of gauge theory of gravity. It is found that, when an inertial reference system is transformed into a uniformly accelerated system by a local gravitational gauge transformation, a non-trivial gravitational gauge field appears. If there is a mass point in the new reference frame, there will be a non-trivial gravitational force acting on it. The nature and the characteristic of this new force are completely the same as those of the traditional inertial force. This new gravitational force is considered to be the inertial force. Therefore, the nature of inertial force is gravity, which is the basic idea of the equi-valence principle.

According to the Newton’s second law of motion, when there is an external force F acted on a mass point m, the mass point will be uniformly accelerated along the direction of the external force. In this paper, we will study another problem: if we sit in the center of mass system of the mass point m, what will we discover? This question should be answered by an astronaut. When an astronaut sits in a rocket which is launching, what will he discover? He will discover that there is a huge force acting on him which is many times larger than his weight. Where does the extra force come from? What is the nature of the extra force? It is known that there are four kinds of fundamental interactions in nature. They are electromagnetic interaction, weak interaction, strong interaction, and gravitational interaction. Firstly, this force cannot be weak interaction or strong interaction, for these two kinds of interactions are short range. Secondly, this force cannot be electromagnetic force, for the astronaut carries no electric charge. Therefore, the only possibility left is that it is a gravitational force. But this force cannot be explained in either Newton’s classical theory of gravity [

In this paper, the problem of inertial force is studies in the framework of quantum gauge theory of gravity. Quantum Gauge Theory of Gravity (QGTG) is proposed in 2001 [

In the literature [

A simple introduction on the quantum gauge theory of gravity is given in this chapter. Details on this theory can be found in literatures [

, (1)

where

where g is the gravitational coupling constant and matrix G is given by

Its inverse matrix is

Using matrix

Quantum gauge theory of gravity is formulated in absolute space-time [

The field strength of gravitational gauge field is defined by

where

The Lagrangian of the quantum gauge theory of gravity is selected to be

where

Its space-time integration gives out the action of the system

Under gravitational gauge transformations [

The gauge transformation of gravitational gauge field is

Using Equation (1), the above relation can be changed into

where

and

Now, let’s study the Lorentz transformation of

where

The reference after Lorentz transformation is a moving reference, which is denoted by S. The gravitational gauge field

By using (3) and (4), the following two relations can be obtained

For the sake of convenience, we suppose that the Lorentz transformation is along the direction of X axis. In this case, we have

where v is the velocity of the boost, and

Now, let’s study the uniformly accelerated motion. In this case,

In this case, the moving reference S is a uniformly accelerated reference. The gravitational gauge field in the moving reference S is

The field strength of gravitational gauge field is given by (8), that is

It is found that all components of

which corresponds to the following gravito-electric field

So, there is a Newtonian gravitational term in the moving reference S, whose direction is along positive X axis, and whose magnitude is

Through the study in the above chapter, we found that there exists non-trivial gravitational field in the uniformly accelerated reference S. Next, we will calculate the gravitational force on a mass point in the reference S. The gravitational force on a mass point with mass m in the reference S is given by

Suppose that the mass point is moving along the X axis, that is

Please remember that the velocity

Substitute (6), (26), (28) and (31) into (30), we will obtain the gravitational force on the mass point

In the above relation,

In this case,

It means that the mass point rested in the accelerated reference S feels a gravitational force with the magnitudes of

Through discussions in this paper and literature [

If the gravitational gauge transformation is global, the gravitational gauge field

The correspondent of gravitational gauge transformation in classical mechanics is the transformation of a local reference. According to Equation (17), when the state of motion of a local reference is changed, the gravitational gauge field in that local reference will be changed accordingly. In other words, the gravitational gauge field in different local reference will be different, and the corresponding gravitational force on a mass point will also be different. The real gravitational force on a mass point depends on the state of motion of a reference. For example, an object which is at rest on the earth will feel the gravitational force of the earth. But if it is in a local inertial reference, it cannot feel any gravitational force, for the gravitational force of the earth and the vacuum gravitational force cancel each other out. This conclusion is the same as that in general relativity.

What is the classical correspondent of the vacuum gravitational force? According to the discussions in this paper, it is known that the magnitude of vacuum gravitational force is the mass times the acceleration of the reference. The direction of the vacuum gravitational force is just the opposite direction of the acceleration of the reference. These properties are the same as those of the inertial force in classical mechanics. Therefore, the classical correspondent of the vacuum gravitational force is the inertial force in classical mechanics.

Because the classical correspondent of the vacuum gravitational force is the inertial force, the nature of the inertial force is gravity. In other words, inertia and gravity are essentially the same, gravity and inertia are homologous, which is just the core idea of the equivalence principle of the general relativity. It is known that the equivalence principle is a transcendental principle in general relativity. But in quantum gauge theory of gravity, it is obtained through a strict derivation. Essentially speaking, it is only a deduction of the gauge principle.

Through the discussions in this paper, we know that there are two ways to produce gravitational field and gravitational force. One way is that it is produced by a massive object, which is given by classical Newtonian gravity and general relativity. Another way is that it is produced by a transformation, which is a new way to produce gra- vitational field and gravitational force. It is an inevitable outcome of gravitational gauge symmetry.

Wu, N. (2016) The Nature of Inertial and Vacuum Gravitational Field. Journal of Modern Physics, 7, 2126- 2134. http://dx.doi.org/10.4236/jmp.2016.715185