1. Introduction

In classical mechanics, the kinetic energy $K_{\mathrm{cl}}$ of a body with mass $m_0$ is given by the following formula:

Here, the subscript “cl” of $K$ indicates that this is the formula for classical mechanics.

Classical mechanics does not take into account the special theory of relativity (STR), and thus there is no need to distinguish between rest mass and relativistic mass.

However, in Einstein’s STR, the two types of mass must be distinguished.

According to the STR, the following relation holds between the energy and momentum of a body moving in free space ^[1]:

Here, $m_0{c}^2$ is the rest mass energy of the body, and $m{c}^2$ is the relativistic energy.

In the STR, there is the following relationship between rest mass $m_0$ and relativistic mass $m$:

When $\beta$ is extremely small, Formula (3) can be expanded as a power series in $\beta$, as indicated below:

It is known that, if a body is moving at low velocity, Formula (4) can be approximated at high precision using the first two terms. Thus,

The second term on the right side of Formula (5) is the kinetic energy in Newtonian mechanics.

Now, how is the relativistic kinetic energy of a body defined in the STR?

Einstein and Sommerfeld defined the relativistic kinetic energy $K_{\mathrm{re}}$ as follows ^[2]:

The “re” subscript of $K_{\mathrm{re}}$ stands for “relativistic.”

Now, Formula (2) is rewritten as follows:

Comparing Formulas (6) and (7), the relativistic momentum $p_{\mathrm{re}}$ can be defined as follows:

Hence,

The following relation holds due to Formulas (6) and (9):

Based on the above discussion, it was found that the relativistic kinetic energy of a body moving in isolated systems in free space can be described with Formulas (6) and (10).

2. Formula for the Classical Kinetic Energy of a Body, Derived from a Wave Standpoint

According to Maxwell’s electromagnetism, the following relationship holds between the momentum $p$ and energy $E$ of light:

If a photon as a single particle is assumed to have a frequency $\nu$, Einstein concluded it has the following energy ^[3]:

Here, $h$ is the Planck constant. Also Formula (12) can be written as follows using the angular frequency $\omega$:

$\omega$ is defined as follows:

The following formula can be derived from Formulas (11) and (12):

Also, the wavenumber $k$ is defined as follows:

de Broglie applied Formula (15) to matter. In classical physics, the following relation holds between momentum $p$ and kinetic energy $K$:

Here, if Formulas (13), (15), and (16) are used,

Therefore,

The phase velocity $v_{\mathrm{phase}}$ and group velocity $v_{\mathrm{group}}$ of a material wave are defined as follows (in the following, these may be abbreviated as $v_{\mathrm{p}},v_{\mathrm{g}}$):

In light of the above, the phase velocity of the wave is as follows:

Also, the group velocity of the wave is as follows:

Hence, the following relationship holds in classical mechanics:

The velocity of a particle $v$ corresponds to the group velocity of a wave $v_{\mathrm{g}}$.

If Formula (23) is taken into account here, Formula (1) for the kinetic energy of a particle can be written as follows:

Therefore,

This shows that there are two kinds of kinetic energy formulas in classical mechanics. Formula (25) incorporates the wave nature of the body.

Now, if the mass of a moving body increases and a relativistic formula becomes necessary, can that formula be found simply by replacing the rest mass $m_0$ in Formulas (1) and (25) with the relativistic mass $m$, as follows?

This problem is considered in the next section.

3. Formula for the Relativistic Kinetic Energy of a Body, Derived from a Wave Standpoint

The phase velocity $v_{\mathrm{p}}$ of a particle is given by the following formula:

$\lambda$ and $\nu$ are the wavelength and frequency of the particle.

Formula (28) can be written as follows using the relationship of Formulas (12) and (15) (velocity and frequency are easily confused, so caution is necessary):

In standard relativistic de Broglie wave theory, the frequency is defined using total energy $E=m{c}^2$ as $\nu =E/h$, which leads to the superluminal phase velocity $v_{\mathrm{p}}=c^2/v$. In the present formulation, the frequency is associated specifically with the particle’s kinetic energy $K$ via $\nu =K/h$ to investigate the wave-mechanical behavior intrinsic to relativistic kinetic energy rather than rest-mass energy, resulting in the subluminal phase velocity $v_{\mathrm{p}}=K/p$.

Due to the above, the formula for the relativistic kinetic energy of a particle corresponding to Formula (11) is as follows:

The energy of a photon is found as the product of the photon’s momentum and the speed of light. The kinetic energy of a particle, in contrast, is determined by the product of the particle’s momentum and its phase velocity.

Here, the phase velocity of the matter wave is derived with two methods by appropriately combining those formulas.

First, identifying the particle velocity $v$ with the wave group velocity $v_{\mathrm{g}}$ ($v=v_{\mathrm{g}}$), the relativistic momentum is expressed as $p_{\mathrm{re}}=m{v}_{\mathrm{g}}$. Substituting this together with the relation $K_{\mathrm{re}}=p_{\mathrm{re}}^2/(m_0+m)=m^2{v}_{\mathrm{g}}^2/(m_0+m)$ from Formula (10), the ratio $K_{\mathrm{re}}/p_{\mathrm{re}}$ yields:

From this, the relationship of $v_{\mathrm{p}}$ and $v_{\mathrm{g}}$ is as follows:

Incidentally, Formula (32) can be written as follows:

Here, the following formula is obtained by multiplying the left side of Formula (34) by $m{v}_{\mathrm{g}}$ and the right side by $p_{\mathrm{re}}$:

Taking Formula (10) into account, this shows that the following relationship has been proven:

In this paper, the following formula was derived as the formula for relativistic kinetic energy:

Also, the following formula can be derived if Formula (30) is taken into account:

The formula for relativistic kinetic energy has previously been defined as in Formula (6), but in this paper two new formulas have been derived.

If this body is an electron, Formula (39) becomes as follows ^[4]:

Here, $m_{\mathrm{e}}$ is the rest mass of the electron.

Also, from Formula (36),

Taking into account the theory of relativity, the relationship between $v_{\mathrm{p}}$ and $v_{\mathrm{g}}$ becomes Formula (33) instead of Formula (23), and thus Formula (26) does not hold. That is,

Hence, it was found that Formula (27) holds, but Formula (26) does not.

Formula (41) can also be written as follows:

Beyond its mathematical compactness, Formula (39) is physically notable because it expresses relativistic kinetic energy directly in terms of the phase velocity $v_{\mathrm{p}}$ of the matter wave, highlighting wave-particle coupling without depending explicitly on group velocity $v_{\mathrm{g}}$ as in Formula (43).

Also, when the particle moves at low speed, $m\approx m_0$, so the approximation in Formula (39) becomes as follows:

4. Conclusions

The formula for the kinetic energy of a body in classical mechanics is Formula (1), but this paper further derived the following formula:

However, in the case of relativistic kinetic energy it becomes as follows:

If Formula (46) is taken into account, it can be concluded that the natural formula for kinetic energy in classical mechanics is Formula (45) not Formula (1).

It was found that, when the formula for the relativistic kinetic energy of a body is derived from these formulas, the body must be treated as a wave, not a particle.

Previously, it has been thought what determines the kinetic energy of a particle is the mass and velocity of the particle.

However, it was found that what determines the kinetic energy of a particle is the mass of the particle and the group velocity and phase velocity of the particle as a wave.

There are also formulas that include phase velocity only, like Formula (49).

Even if Formulas (39) and (43) are compared, it is concluded that phase velocity is more essential than group velocity as a factor determining kinetic energy.

In today’s standard physics, the de Broglie wave is widely understood within a probabilistic interpretation, and the wave function does not describe a classical physical wave. The author acknowledges this standard framework. While the mathematical formulations derived in this work show that relativistic kinetic energy can be expressed through wave variables such as phase velocity and group velocity, this mathematical representation alone does not constitute experimental proof of the physical ontology of matter waves. Nonetheless, the fact that wave parameters can consistently describe relativistic kinetic energy provides theoretical motivation for re-examining the physical interpretation and implications of matter waves ^{[5, 6]}.

Acknowledgments

I would like to express my thanks to the staff at ACN Translation Services for their translation assistance.