1. Introduction

Let us consider that we have a positive charge located in the center of a inertial reference frame and a negative charge located in the center of a non-inertial reference frame Then, let us imagine that the non-inertial reference frame is moving with the acceleration in the (x) direction with respect to the inertial reference frame. Also, let us assume that the accelerated charge is moving such that the bilinear form

(1.1)

is an invariant with respect to the following non-linear transformations

(1.2)

where, So, we try to find the functions such that

(1.3)

First we replace the coefficients as follows

(1.4)

Then, substituting the transformations (1.2) into equation (1.3), we get

(1.5)

So, we must have and the transformations (1.2) become

(1.6)

Also, we can admit the following inverse transformations

(1.7)

where

Differentiating now the direct transformations (1.6), we obtain

(1.8)

(1.9)

Differentiating also the inverse transformations (1.7), we obtain

(1.10)

(1.11)

In order to find the physical semnification of the function, we observe that, for the origin of the non-inertial reference frame, the equation (1.10) becomes

(1.12)

From this, we can deduce the derivative of the position with respect to time

(1.13)

which signifies the velocity of non-inertial reference frame with respect to the inertial reference frame. Also, in order to find the physical semnification of the function we observe that, for the origin of the inertial reference frame, the equation (1.8) becomes

(1.14)

From this, we can deduce also the derivative of the position with respect to time

(1.15)

which signifies the velocity of the inertial reference frame with respect to the non-inertial reference frame. But, according to the motion relativity principle, we must have

(1.16)

Therefore, we can admit the following identity

(1.17)

2. The Wave Function

Further on, we can associate a de Broglie wave with the accelerated charge. The wave must be stationary with respect to the non-inertial frame. Therefore, we can introduce the following wave function for the accelerated charge

(2.1)

where is given by the equation (1.7). Thus, we can rewrite this function with respect to the inertial reference frame as follows

(2.2)

Taking the partial time derivative of this function, we get

(2.3)

Now, we can introduce by definition the Hamiltonian operator

(2.4)

According to the equation (2.3), for the point we get

(2.5)

Therefore, we can deduce from here the following Hamiltonian function

(2.6)

where is the rest energy of the accelerated charge

(2.7)

and is the charge rest mass. Taking now the partial derivative of the function we get

(2.8)

Further on, we can introduce by definition the linear momentum

(2.9)

So, according to the equation (2.8), for the point we can write

(2.10)

Thus, we obtain the following expression for the linear momentum of the accelerated charge

(2.11)

Substituting now the equation (1.13) into the equation (2.6), we get the well known formula

(2.12)

where is the charge moving mass

(2.13)

Also, substituting the equation (1.13) into the equation (2.11), we get the well known formula

(2.14)

3. The Hamilton’s Equations

According to the principles of analytical mechanics, we can introduce now the Lagrange function

(3.1)

and the linear momentum as the derivative of lagrange function with respect to

(3.2)

where, according to the equation (1.13), we can consider

(3.3)

Let us consider also the action functional

(3.4)

where are constants. We assume that the action attains a local minimum at and is an arbitrary function that has at least one derivative and vanishes at the endpoints So, for any number close to zero, we can write

(3.5)

Thus, if the functional has a minimum for the function has a minimum at

(3.6)

where, according to the equation (3.3), and are functions of of the form

(3.7)

(3.8)

Taking the total derivative of we get

(3.9)

But, according to the equations (3.7) and (3.8), we have

(3.10)

(3.11)

Therefore, we get

(3.12)

According to the fundamental lemma of calculus of variations, we get

(3.13)

In consequence, we can introduce now the Hamiltonian function as

(3.14)

Differentiating now this function, according to the equation (3.13), we get the following equation

(3.15)

Comparing this expression with the mathematical expression

(3.16)

We get the following Hamilton equations

(3.17)

(3.18)

4. The Schrodinger Equation

Further on, we consider a function which describes a physical quantity of the accelerated charge. Taking the total time derivative of the function we obtain

(4.1)

According to the equations (3.17) and (3.18), we can rewrite this formula as follows

(4.2)

where is the classical Poisson bracket

(4.3)

Let us write the equation (4.2) under the following form

(4.4)

where is an operator whose expression is given by

(4.5)

Taking into consideration the analogy between quantum mechanics and analytical mechanics, we can introduce an operator corresponding to the physical quantity F. According to the equation (4.4), the corresponding equation for this operator must be

(4.6)

where is the quantum Poisson bracket

(4.7)

According to the equation (4.6), we can define the time derivative of the expectation value of the observable as follows

(4.8)

where is the mean value (expectation value) of the observable Also, the law that describes the time evolution of a particle must be of the form

(4.9)

In this equation, is the wave function of a quantum system, in the representation. According to this representation, the operators and are given by the expressions

(4.10)

Therefore, in the expression (4.5), we can make the replacing

(4.11)

which leads us to the following expression of the operator in the representation

(4.12)

But we seek the Schrodinger equation (4.9) in the representation. For this, we must introduce the operators

(4.13)

and the canonical commutation relation

(4.14)

According to this relation, in the expression (4.12) we can make the replacing

(4.15)

which leads us to the following expression of the operator in the representation

(4.16)

Therefore, we can write now the following Schrodinger equation

(4.17)

where the new Hamiltonian operator for the accelerated charge is given by

(4.18)

The additional term is of complex nature and its real part signify the potential energy of the field of force in which the particle is moving.

5. The Damped Oscillator

Let us consider that the accelerated charge is bound to the fixed charge by an elastic force of the form

(5.1)

Taking the total time derivative of the equation (2.6), we can introduce, according to the special relativity theory, the following equation

(5.2)

where is the resultant force which acts on the accelerating charge

(5.3)

This force must be the sum between the elastic force and the radiative reaction force

(5.4)

Therefore, for non-relativistic velocities, we get the equation of motion

(5.5)

We know that this equation is applicable only to the extent that the radiative reaction force is small compared with the elastic force

This condition leads us to the following equation of motion

(5.6)

where

(5.7)

Thus, the approximate solution of the equation (5.6) can be written as

(5.8)

where is the angular frequency of the damped oscillator

(5.9)

Also, from the equation (5.2), for non-relativistic velocities, we can now deduce a new differential equation

(5.10)

The integration of this equation leads us to the solution

(5.11)

where is given by the expression

(5.12)

But, according to the equation (1.13), we must have the condition

(5.13)

Taking the total time derivative of this equation, we get

Now, we can replace

Therefore, we get the equation

which is the same equation (5.6). Substituting this result into the equation (5.10), we can write the approximate equation

(5.14)

The solution of this equation is given by the expression

(5.15)

which results from the equation (1.13). Taking now the total time derivative of the displacement (5.8), we get

(5.16)

This expression can be written as a function of the displacement function under the following form

(5.17)

where the function is given by the expression

(5.18)

In this way, the velocity becomes a function of the coordinate , because here the displacement is considered as a variable which no longer depends on time. Consequently, we may admit for a solution of the form

(5.19)

6. The Damped Quantum Oscillator

Substituting now the solution (5.17) into the equation (4.18), we get, for non-relativistic velocities

(6.1)

For the Hamiltonian operator of the oscillator, we take the expression

(6.2)

Admitting for the wave function a solution of the form after the separation of variables, we get the following two equations

(6.3)

(6.4)

where the constant is the separation constant, and signifies the total energy of the quantum oscillator. Thus, after a rearrangement of the terms, the first equation can be rewritten as follows

(6.5)

Further on, we can introduce a dimensionless variable

such that the equation becomes

(6.6)

in which we have introduced the notations

(6.7)

(6.8)

(6.9)

Now, we can take for a solution of the form

(6.10)

which generates the differential equation for the unknown function

(6.11)

The constant can be determined from the equation

(6.12)

In order to get a finite solution for the function we choose the root

(6.13)

where we have introduced the notation

(6.14)

Therefore, the equation (6.11) becomes

(6.15)

We can introduce now a power series expansion as solution

(6.16)

Substituting this expansion into the differential equation (6.15), we get

(6.17)

which leads us to the recursion relation

(6.18)

Because the series must be finite, there exists some such that when the numerator will be zero. Therefore, we get

(6.19)

Substituting here the formula (6.8), we obtain the following complex expression for the total energy of the quantum oscillator

(6.20)

where are the quantized energies of the quantum oscillator

(6.21)

Substituting now the constant into the expression (6.10), we get

(6.22)

Also, substituting the equation (6.19) into the equation (6.15), we get for the unknown function the equation

(6.23)

If we introduce now a new dimensionless variable

(6.24)

the equation (6.23) becomes

(6.25)

where are the Hermite polynomials. So, and the solution can be written as

(6.26)

where are the multiplication factors, whose expression can be determined by the normalization condition of the wave function.

For the second equation (6.4), we have the solution

(6.27)

Substituting here the solution (6.20), we get the expression

(6.28)

which describe the time evolution of the wave function for the accelerated charge.

7. The Metric of Space-Time

Let us write now the “distance” between two infinitesimally close events, into the inertial reference frame

(7.1)

Substituting here the equations (1.8) and (1.9), we get

(7.2)

where the components of the metric tensor are

(7.3)

(7.4)

(7.5)

But, according to the equations (1.17) and (5.11), we have

(7.6)

Then, according to the transformations (1.6), we can impose

(7.7)

(7.8)

where

(7.9)

Substituting now these expressions into the equation (7.6), we get

(7.10)

where

(7.11)

Taking now the partial derivative of we obtain

(7.12)

Also, taking the partial derivative of we obtain

(7.13)

But, according to the transformations (7.7) and (7.8), we can write

(7.14)

(7.15)

(7.16)

Substituting these expressions into the equations (7.12) and (7.13), we get

(7.17)

(7.18)

where the velocity is given by the expression (5.17), which no longer can be written as a function of the coordinates So, in this particular case, the metric of space-time (7.2) becomes a Finsler metric, and the space becomes a Finsler space.

8. Conclusions

Intuitively, the non-uniformly accelerated linear motion of a oscillator must be equivalent to a uniform but variable gravitational field. However, the gravitational field described by the metric of space-time (7.2), is a non-uniform field. This is due to the particular mode of writing of the velocity of a particle as a function of the coordinates and the function depending on the same coordinates, by means of the velocity. Also, from the equation (3.17), we can observe that the term corresponds at the half of the radiative reaction force

Therefore, the function which determines the Lorentz non-linear transformations, intervenes in the determination of the radiative reaction force of the EM field and, also, in the determination of the gravitational field which appears in the non-inertial reference frame. Indeed, if we impose now we get

so, both, the radiative reaction force and the gravitational field, vanish. On the other hand, the radiative reaction force vanishes again when the oscillator is a neutral particle, but the gravitational field does not vanish, because is different from zero. This suggests that the EM energy carried by the EM radiation contributes to the energy of the gravitational field. Also we can conclude that the field which appears as an EM field into the inertial reference frame, appears as a gravitational field into the non-inertial reference frame.