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Introducing new equation for total energy, potential energy, and kinetic energy in the
multidimensional spacetime
Bahram Kalhor
1
, Farzaneh Mehrparvar1, Behnam Kalhor1
Abstract
The paper introduces the energy equation in a multi-dimensional spacetime. Firstly, we use a new
relation between kinetic energy and potential energy. Secondly, we obtain total energy based on
potential energy and kinetic energy. Thirdly, we present the relationship between the momentum
and the kinetic energy, and Finally, we compare the proposed equation with the mass-energy and
the energy-momentum equation.
Introduction
The total energy and potential energy have been related by Paul Dirac in 1928. Its equation is given
by +V where E is the total energy, p is the momentum, c is the speed of
light, is the rest mass, and the V is the potential energy [1]. On the other hand, In the past
century, physicists have used the idea of defining kinetic energy. The root of the kinetic in the
Greek is “κίνησις“[2] that is equal to motion. Newtonian kinetic energy is defined based on the
mass and velocity (
) [3-8].
In this paper, we use a new definition of the kinetic energy [9] and introduce a new equation for
describing the total energy of the particles. The new equation gives us the energy of the n-
dimensional particle when it is moving with a certain velocity. The kinetic energy is perpendicular
to the potential energy.
Kinetic energy and Potential energy
The popular equation which is used in the past century for describing the kinetic energy is given
by:
(1)
where is the kinetic energy, m is the mass of the particle, and v is the velocity of the particle.
The equation (1) approximates the relativistic kinetic energy when the velocity of the particle is
too less than the speed of light, while we use a new equation that describes the kinetic energy based
on the potential energy.
1
Independent researcher form Alborz, IRAN
Corresponding author. Email: Kalhor_bahram@yahoo.com
In the past century, for obtaining kinetic energy of the particles, physicists have used the relativistic
energy minuses the rest energy. The equation is given by:
(2)
where E is the relativistic energy, is the rest energy, and is the kinetic energy.
According to the mass-energy equivalence, the rest mass for a three-dimensional particle is given
by:
(3)
where is the mass of the particle and c is the speed of light, hence:
In this paper, in each dimension the kinetic energy is equal to the potential energy at the beginning
of the latest dimension multiplied by the velocity of the particle [10]. The equation is given by:
(4)
where is the kinetic energy of the particle in the dimension number n ( latest dimension),
is the potential energy of the particle at the beginning of the dimension n, n is the number of the
dimension, and v is the velocity of the particle.
At the speed of light, the kinetic energy of the particle increases to c times of its potential energy.
After reaching the speed of light, kinetic energy will be converted to potential energy. Hence, the
kinetic energy is the energy of motion in the latest dimension, while the rest energy is the energy
of the particle when its velocity in the latest dimension is equal to zero. Converting the kinetic
energy does not mean increasing the mass,
The potential energy in each dimension is given by [9]:
(5)
Where is the potential energy, and n is the number of dimensions.
(6)
Using (4)
(7)
or
(8)
Total energy
The total energy is the sum of the kinetic energy and the potential energy. As we mentioned in the
[9-10] kinetic energy is perpendicular to the potential energy, and we should use vectorially
summation to calculate total energy, the equation is given by:
(9)
is perpendicular to the , using (4)
(10)
Or using (5), (7) and (9)
(11)
Equations (10) and (11) show that at the beginning of each dimension when the speed of the
particle is equal to zero, the total energy of the particle is equal to the potential energy. Also, at the
end of the dimension when the speed of the particle reaches the speed of light, the total energy of
the particle is equal to (c+1) times of its potential energy at the beginning of the dimension.
By combining equations (5) and (10) we can relate the total energy of the particle to its mass, the
equation is given by:
(12)
or
(13)
For instance, by using equation (5), the potential energy, and the total energy of a three-
dimensional particle is given by:
+ mc + m (14)
or
and
(15)
or
(+ mc + m) (16)
Equation (16) is the best definition of the total energy for a three-dimensional particle. On the other
hand, , hence we can make a new approximation of the energy equation:
(17)
Obtaining multidimensional energy-momentum
Angle between the kinetic energy and potential energy is equal to
, hence we can write the
equation (9) in this form:
(18)
or
(19)
Using (5) , (7)
(20)
or
(21)
(22)
Hence
(23)
Using (23) when n=3 gives three energy-momentum equation:
(24)
or
(25)
Conclusion
We proposed a new equation for relating the total energy, potential energy, and kinetic energy.
The energy equation ( or ) implies that the kinetic
energy is the energy of motion in the latest dimension, while it is perpendicular to potential
energy. Also, the potential energy at the beginning of each dimension is almost equal to the mass
of the particle multiplied by the where n is the number of the dimension. Mass does not
increase by increasing the velocity. After reaching the speed of light, the particle starts
oscillating in a new dimension while saving its velocity and previous path of the movement.
Hence, particle combines its new direction and previous path. Equation (25) is the multi-
dimensional energy-momentum equation in the three-dimensional real space (
), and matching with [11], while contradicting with relativistic energy-
momentum equation.
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