_{1}

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When we study Lorentz transformation in the framework of quantum gauge theory of gravity, we will find that the vacuum gravitational gauge field will be changed under gravitational gauge transformation, which will change the structure of the physical space-time and cause clock dilation effect. The study in this paper provides us with new insights to understand the essential and intrinsic relation between special relativity and general relativity. It provides us with a new way to unify special relativity and general relativity.

It is known that, because of the negative results from Michelson-Morley interference experiment [

Besides in special relativity, clock dilation effect and ruler contraction effect also exist in general relativity [

Quantum Gauge Theory of Gravity (QGTG) is proposed in 2001 [

In this paper, we first use the language of gauge transformation to formulate Lorentz transformation, and study the change of the structure of physical space-time under this gauge transformation. It will help us to understand the physics behind Lorentz transformation and the nature of Lorentz symmetry.

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

where

The gravitational gauge covariant derivative is given by

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 space-time coordinates is

The gauge transformation of gravitational gauge field is

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

where

and

The fundamental theory for gravitational interactions can be formulated in two completely different pictures [

Two pictures of gravity have completely different transcendental principles, completely different basic physical notions, and completely different mathematical treatment. But for problems of classical gravity, theories of two pictures give out the same theoretical predictions. In other word, for problems of classical gravity, two pictures of gravity are finally equivalent to each other.

In physical picture of gravity, space-time is always flat, so we call it absolute space-time. A coordinate system which is set up in absolute space-time is denoted by

The above equation gives out the relation between two space-times.

Now, let’s study Lorentz transformation. Suppose that we are in absolute space-time. The coordinate system is

where

where

Next, we will study the above Lorentz transformation in the framework of quantum gauge theory of gravity. Gravitational gauge transformation of space-time coordinate is given by Equation (12). Compare Equation (12) with (18), we obtain

Therefore, the gravitational gauge transformation parameter

It could be seen that the transformation parameter

The gravitational gauge transformation of gravitational gauge field

we can change Equation (14) into the following form

Using Equations (16) and (22), the above relation can be changed into

In the above relation, the matrix

It satisfies the following relation

Using above relation, we can change Equation (25) into the following form

Before Lorentz transformation, we are in absolute space-time. Suppose that there is no matter field, and the gravitational field vanishes in all points of space-time. That is

Then, relation (28) is changed into

From above relation, we could see that, after Lorentz transformation, the gravitational field no longer vanishes. There exists constant gravitational field in space-time. Because the gravitational field after Lorentz transformation is constant, its space-time derivative vanishes and the field strength of gravitational gauge field also vanishes. Therefore, there is no gravity in the reference frame after Lorentz transformation, and the reference frame after Lorentz transformation is still an inertial reference frame as expected.

A basic idea of modern theory on gravity is that the time interval between ticks of a clock and the length of a ruler are all changed by the classical gravitational field, and the structure of physical space-time is also affected by the classical gravitational field. Before Lorentz transformation, the gravitational gauge field vanishes everywhere. But after Lorentz transformation, there exists non-trivial gravitational gauge field which is given by Equation (30). Its explicit form is

Though all matrix elements in gravitational gauge field

Next, let’s study its physical implications. The matrix G which is defined by Equation (3) has the following explicit form

Its inverse matrix

Next, we perform our study in physical space-time. Because the gravitational gauge field vanishes before Lorentz transformation, according to Equation (17), the absolute space-time and the physical space-time are the same. Therefore, before Lorentz transformation, the space-time interval of a physical event can be denoted by

The above relation is the same as that given by Lorentz transformation (18). The above relation is deduced from the point of view of gravitational gauge transformation. It is a relation that is obtained from the change of the space-time structure after gravitational gauge transformation.

Supposed that there is a clock at rest in the reference frame

This is the clock dilation effect given by the change of the gravitational field.

Suppose that there is a ruler that is at rest in the reference frame

It is the ruler contraction effect caused by the change of the gravitational gauge field. Relations (35) and (36) are familiar results in special relativity. But here, they are deduced from the viewpoint that the time interval between ticks of a clock and the length of a ruler are all changed when the classical gravitational field is changed.

In this paper, the transformation law of vacuum gravitational gauge field and the change of space-time structure under gravitational gauge transformation are studied. It is found that, when the global Lorentz transformation is studied using the method of gravitational gauge transformation, the vacuum gravitational gauge field is changed under the transformation, which will cause the change of space-time structure. The clock dilation effect and the ruler contraction effect are results of the change of space-time structure. It is known that, in special relativity, the Lorentz transformation is only a mathematical transformation of the reference system, or say that it is only a transformation of the moving state of the observer. The change of mathematical parameters of a theory generally can not affect our clock and ruler. In other words, when we change the mathematical parameters of a theory that describe a physical event, the time interval between ticks of a clock and the length of a ruler generally should not be changed accordingly. For a long time, we cannot understand the physical mechanism that causes the change of space-time structure under such mathematical transformation. The goal of this paper is to study such physical mechanism, and we found that the physical mechanism that causes the change of space-time structure under Lorentz transformation is that the change of classical gravitational field causes the change of space-time structure, which is familiar for us in general relativity. In the traditional theories, the picture of the change of the space-time structure in special relativity is completely different from that in general relativity. In special relativity, it is traditionally considered to be an effect of kinematics, and in general relativity, it is considered to be a result of gravity. From the study of this paper, we found that two pictures can be unified, and they are essentially the same. In other words, the clock dilation effect and the ruler contraction effect in special relativity are also effects of gravity. What that the space-time structure is changed by classical gravitational field is a more fundamental law in physics.

It is known that the generalization of special relativity is general relativity, which is a theory of gravity. But, why will the generalization of special relativity which is a kinematical theory inevitably lead to a fundamental theory on gravity? It is hard to understand the physical nature and the inevitability of such generalization just from the point of view of symmetry. Now we know that the physical mechanism that hides behind the Lorentz transformation is gravity and the influence of gravity to space-time structure. It provides us with new insights to understand the essential physical relation between special relativity and general relativity. From physical point of view, the nature of the generalization from special relativity to general relativity is a generalization of gravity, or say that, it is a generalization from the theory of uniform gravitational field to the theory of arbitrary gravitational field.

It is generally believed that the classical gravitational field in inertial reference system should vanish. It is known that, if the initial reference system is inertial, after Lorentz transformation, it is still inertial. According to Equation (31), the classical gravitational field in inertial reference system can be non-zero, and the constant gravitational field can have non-trivial influence to the space-time structure.

The gravitational gauge field after gravitational gauge transformation is given by Equation (31). Using relations (5) and (6), we could calculate the metric of physical space-time. We find that the metric of physical space- time after Lorentz transformation is still Minkowski metric. So, the gravitational gauge field given by Equation (31) is a solution of the field equation of gravitational gauge field. By the way, we should state that results in this paper cannot be obtained in the traditional formulation of general relativity; for the gravitational field

Equation (17) gives out important relation between two space-times. It is known that the structure of absolute space-time is fixed; it cannot be changed by any man-made machine. But with the guide of Equation (17), it is possible for us to change local structure of physical space-time, which will have far reaching influence on human kinds; for example, we can make a machine that can essentially prolong human being’s life by physical method. Details on this topic can be found in [