_{1}

^{*}

In this article, a novel speculative method is used to derive the relativistic mechanic that governs the motion of the vibrating string within the compactified-dimensions spacetime. This mechanic claims that the relativistic mechanic of the special relativity should be only valid for the motion within the familiar four-dimensional spacetime. However, our novel mechanic is valid for the motion within the compactified-dimensions spacetime predicted by the string theory. The equations of this new mechanic show that the vibrating string can move within the compactified dimensions in a speed that is faster than light. It is also shown that this new relativistic mechanic goes to the classical Newtonian mechanic whenever the speed of the vibrating string is much less than the speed of light. Since the proposed mechanic does not prohibit the existence faster than light motion, it may uncover some of the mysteries regarding the string theory, such as the existence of tachyon and time travel. The main goal of this paper is to show that the motion within the compactified-dimensions spacetime obeys a different relativistic mechanic that will provide a startling and revolutionary perspective on the universe and answer some of the fundamental questions posed in the modern physics.

Before 1905, physicists used the Newtonian mechanic to describe the motion of any object without considering any upper limit on the speed. This mechanic was formulated by observing and describing the motion of such objects. This formalism is very successful in describing a wide range of phenomena that occur at low speeds. However, it fails to describe properly the motion of objects whose speeds approach that of light [

Although the special relativity gives a better vision for the universe and spacetime, we still believe that there is a limit for the applicability of this theory. In our point of view, the relativistic mechanic of the special relativity can be applicable to the motion within the familiar four-dimensional spacetime. However, the motion within the compactified-dimensions spacetime predicted by the string theory requires another relativistic mechanic that needs to be discovered. Therefore, a novel speculative method is used to derive this relativistic mechanic. This method is based on the suggestion stated by the American physicist Richard Feynman in 1994. He suggested that a positron could be described mathematically as an electron that was traveling backward in time [

The two famous postulates of the special theory of relativity lead to the following relativistic relationships,

where m is the relativistic mass, m_{o} is the rest mass, u is the velocity, c is the speed of light, E is the relativistic energy, K is the relativistic kinetic energy, and p is the relativistic momentum. As it can be easily noticed, these equations give imaginary values whenever u is greater than c [

The special relativity equations can be applied to any particle moving in spacetime with four dimensions without considering the motion of its inner structure (i.e. the motion of its string(s)) that is happening in spacetime with compactified dimensions. For this inner motion of the particle, we will, speculatively, derive different relativistic mechanic that give no imaginary mass for the superluminal motion which might give a new view for the concept of tachyon.

Experiments have proven the correctness of the special theory of relativity. The equations of this theory can be applied to any accelerated particle that is composed of one vibrating string (e.g. the electron) or more than one vibrating string (e.g. the proton). This means that the motion within the four-dimensional spacetime has an individual effect on every single vibrating string that is moving within the compactified-dimensions spacetime.

Now, in order to derive the relationship between the relativistic mass and the velocity of the vibrating string, we will consider both the electron and its anti-particle, the positron. In 1949, the American physicist Richard Feynman showed that a positron could be described mathematically as an electron that is traveling backward in time [

where m is the relativistic mass,

Equation (5) gives the relativistic mass in terms of the rest mass and velocity within the compactified-dimen- sions spacetime. However, Equation (1) gives the relativistic mass in terms of the rest mass and velocity within the four-dimensional spacetime. Equation (5) also shows that the vibrating string can travel faster than light within the compactified-dimensions spacetime. However, as we will see from Equations (7) and (12), this superluminal motion cannot be noticed in the four dimensional spacetime.

Assuming in the rest case of the electron (i.e. u = 0 and m = m_{o}), its vibrating string is moving within the compactified dimensions with a velocity of v_{1}, then using Equation (5) we can write,

Substituting Equations (5) and (6) into (1) we find the relationship between u and v as following,

Interestingly, if we rewrite Equation (5) in the form

then we expand the function under the square root into a Taylor’s series, it becomes

which clearly appears to be very similar to Equation (1).

From Equations (2) and (5), we find that the relativistic energy represented by a vibrating string is,

which shows that the energy still equals mass times the speed of light squared. This equation gives the total relativistic energy in terms of the rest mass and velocity within the compactified-dimensions spacetime. However, Equation (2) gives the total relativistic energy in terms of the rest mass and velocity within the four-dimensional spacetime.

Consequently, the relativistic kinetic energy will be,

or

If we expand Equation (10) into a Taylor’s series, we obtain,

which goes to the classical Newtonian expression for

If we substitute Equations (5) and (7) into Equation (4), it becomes

This equation represents the relativistic momentum in u-direction (i.e. the four-dimensional spacetime momentum). However, the relativistic momentum for the vibrating string in v-direction (i.e. the compactified-di- mensions spacetime momentum) can be obtained by letting v_{1} = 0. Therefore, by substituting in Equation (11) we obtain

This equation is plotted in

which also goes to the classical Newtonian expression for

Equation (12) can also be obtained by the classical Newtonian mechanic with one modification. From the classical mechanic we have

thus

^{1}This equation shows that whatever the velocity of the vibrating string is within the compactified-dimensions spacetime, it will never exceed the speed of light within the four-dimensional spacetime. In fact, if we expand Equation (14) into a Taylor’s series, then we will notice that both velocities are equal at

Since we are dealing with different type of spacetime, we should replace the velocity u here with the velocity v. How can we do this? In fact, we can just easily use Equation (7) after letting v_{1} = 0. This will make Equation (7) looks like

^{1}

This equation relates the velocity of the vibrating string (v) within the compactified-dimensions spacetime to the total velocity of the vibrating string within the four-dimensional spacetime. Now, by substituting Equations (10) and (14) into (13), we obtain

by performing the integration we obtain

which is Equation (12).

Also, from Equations (9) and (12) we can obtain the following relationship,

Interestingly, the same expression can be obtained by using the mechanic of the special relativity. This expression suggests that a particle may have energy and momentum even if it has no rest mass. Thus when

By differentiating the momentum Equation (12) with respect to time, we will obtain the force acting on the moving vibrating string

Since the string theory seeks to unify the four fundamental forces in nature, we believe that this equation could carry very important information regarding this unification.

The novelty of the new speculated relativistic mechanic lies in its correctness, consistency and simplicity. This mechanic goes to the Newtonian mechanic whenever the speed is much less than the speed of light. It also shows that there is an infinite number of velocity ranges. However, every vibrating string is trapped within a certain range of velocity and it cannot escape this range. According to this mechanic, the concept of the imaginary mass is physically unrealistic. ^{ }

According to our theory, the two different spacetimes are directly related to each other. For example, if we consider that the vibrating string of the electron is trapped, for instance, within the first range (0 - (π/2)c), then according to this theory, as we are trying to accelerate this electron within the four-dimensional spacetime to make it reach the speed of light, we actually at the same time are trying to accelerate its vibrating string within the compactified-dimensions spacetime to make it reach the speed of (π/2)c.

To this end, we strongly believe that our proposed mechanic can provide the string theory with a strong evidence of its correctness.

This paper suggests that the motion within the compactified-dimensions spacetime predicted by the string theory requires a different relativistic mechanic that is different than that presented by the special theory of relativity. A speculative method is used to derive this relativistic mechanic. It is, accordingly, shown that the vibrating strings of a particle (or any matter) can move faster than light within the compactified-dimensions spacetime. In our point of view, this new mechanic could be the road into more understanding of the laws of nature.