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Is it possible to obtain multi-dimensional energy-momentum equation by using Minkowski
spacetime?
Bahram Kalhor
1
, Farzaneh Mehrparvar1 , Behnam Kalhor1
Abstract
We use Generalized Minkowski spacetime and obtain a multi-dimensional energy equation in the
spacetimes. We use potential energy and total energy as two components in the Makowski
spacetime. We present a new multi-dimensional energy equation that relates the total energy of the
multi-dimensional object to its rest mass. The multi-dimensional energy equation shows that at the
speed of light, the kinetic energy of the particle reaches the potential energy multiplied by the
speed of light. At the speed of light, the kinetic energy will be converted to the potential energy,
hence the potential energy in the higher dimension is c times of the potential energy in the previous
dimension.
Introduction
Physicists are using the Minkowski spacetime to add time as a 4th dimension [1]. The core of the
Minkowski spacetime is Lorentz transforms [2-6]. The Lorentz transform presents a mathematical
view for comparing or transforming the mathematical and physical parameters according to two
different observers in two different frames [7-8].
In the Makowski spacetime, different components could be chosen to extract their relation in the
spacetime [9]. If we consider the energy and the momentum as two components, the energy-
momentum equation would be obtained [10]. Einstein has formulated special relativity by using
the Lorentz transform that is the core of the Minkowski four-vectors [11-23]. In the special
relativity, Minkowski four-vectors has used for relating energy and momentum in 4-dimensional
spacetime [24-26].
In this paper, we use the potential energy and the total energy as two components in the Makowski
spacetime and extract the multi-dimensional energy-momentum equation. We use Generalized
Minkowski spacetime and obtain energy equation in different dimensions. We present a novel
multi-dimensional energy equation that relates the total energy to the potential energy in different
dimensions. In the multi-dimensional energy equation, the total energy of the object relates to the
rest mass and velocity of the object in the latest dimension.
We show that at the speed of light, the kinetic energy increases and will be equal to the potential
energy multiplied by the speed of light, hence the total energy of the object at the speed of light is
(c+1) times of its potential energy. At the speed of light, the whole kinetic energy will be converted
to potential energy. At the speed of light, increasing the velocity in the current dimension will be
1
Independent researcher form Alborz, IRAN
Corresponding author. Email: Kalhor_bahram@yahoo.com
stopped, so then oscillating in the new dimension starts. We show that the potential energy at the
new dimension is c times of the potential energy in the previous dimension.
Minkowski spacetime
Minkowski spacetime provides a four-dimensional spacetime by combining three-dimensional
real space and time. Depends on what components have chosen, the relation between components
could be extracted. In physics, one of the most famous usages of the Minkowski spacetime is
extracting the energy-momentum equation. By using the energy and momentum of an object and
using time as a new dimension, we can extract the energy-momentum equation.
In the Minkowski four-vectors, a three-dimensional momentum generalize into spacetime
. According to which convention we use change. If we use, then
while if we use ,
where E is the energy of the object and is the coordinate
convention.
In the Minkowski four-vectors, the inner product of the momentum in the space-time is given by:
(1)
Where is the Minkowski metric, is the inner product of the momentum in the three-
dimensional space, and is the momentum of the object in the spacetime.
also, =
while
where is the rest mass, E is the energy of the object, and c is the speed of light.
hence,
or
Energy equation in 4 dimensional Minkowski spacetime
In this paper, we use the potential energy and obtain the total energy in the spacetime. Minkowski
spacetime equation is given by:
(2)
where is the potential energy, is the inner product of the potential energy in the three-
dimensional space, v is the velocity of the object, and is the total energy of the object in the
spacetime.
Hence, (3)
where
and
hence, (4)
or (5)
so ) (6)
(7)
The equation (7) shows the relation between the total energy and the potential energy in the
spacetime.
At the speed of light (v=c) all the kinetic energy will be converted to the potential energy, hence
(8)
Where n is the number of the current dimension and n+1 implies to next dimension.
using (6) ) (9)
,hence
so (10)
using (8) (11)
The equation (10) shows that the total energy of the particle at the speed of light is equal to the c
time of the potential energy of the object when the velocity is equal to zero. Also, the equation
(11) shows that at the speed of light the potential energy in the next dimension is equal to c times
of the potential energy of the previous dimension.
Generalized Minkowski spacetime
Minkowski spacetime is not restricted to three real dimensions and time dimension. We can use it
in n-dimensional spacetime () in which we extract relation between the two input
components in a (n-1) real space dimensions and time dimension. By using the potential energy
and the total energy as the two components in the real space, the equation is given by:
(12)
where is the potential energy of the (n-1)-dimensional object, is the inner product of
the potential energy in the (n-1)-dimensional real space, v is the velocity of the object, and is
the total energy of the object in the n-dimensional spacetime, hence:
(13)
The key point of the equation (13) is the quantity of the. According to equation (11), the potential
energies in different dimensions have relationship, hence by knowing the amount of the potential
energy in the one dimension all the potential energies of the all dimensions obtainable. The rest
mass () is the potential energy in the two-dimensional space time (one dimensional particle),
hence we would expected that obtain new equation that relates the potential energy of the n
dimensional object to the rest mass. Before presenting multi-dimensional energy equation we
extract energy equation for 2,3, and 4-dimension spacetime and compare them.
Two-dimensional Minkowski
In the two-dimensional Minkowski spacetime (n=2) where the spacetime contains one real space
dimension and time dimension, the equation is given by:
(14)
where is the potential energy of the one-dimensional particle, is the inner product of
the potential energy in the one-dimensional real space, v is the velocity of the object, and is the
total energy of the object in the two-dimensional spacetime.
Hence, (15)
the potential energy of the one-dimensional mass is equivalence to its mass, hence
(16)
and (17)
using (15), (16), (17)
or
using (16) (18)
The equation (18) is the energy equation that is extracted from the two-dimensional Minkowski
spacetime. The equation (18) shows that when velocity of the particle is equal to zero, the total
energy of the particle is equal to the potential energy, while the potential energy of the particle is
equal to the rest mass.
At the speed of light where we can simplify the equation (18).
hence (19)
and
hence (20)
using (16)
hence (21)
and (22)
Three-dimensional Minkowski
In the three-dimensional Minkowski spacetime (n =3), we have a two-dimensional real space and
time dimension, the equation is given by:
(23)
where is the potential energy of the two-dimensional object, is the inner product of
the potential energy in the two-dimensional space, v is the velocity of the object, and is the total
energy of the object in the three-dimensional spacetime.
Hence, (24)
According to the equation (22), the potential energy of the two-dimensional mass is equivalence
to its mass multiplied by the speed of light, hence
(25)
using (3)
using (25)
hence
or
(26)
(27)
(28)
(29)
The equation (28) and (29) like the equation (18) describes the total energy based on the rest mass.
According to these equations the total energy of the one two-dimensional particle with velocity v
is almost c times of the total energy of the one-dimensional object with the same rest mass and
same velocity.
At the speed of light where , we can change equation (29) to:
(30)
hence
where
so (31)
using (22)
hence (32)
using (22) (33)
Four-dimensional Minkowski
In the four-dimensional Minkowski spacetime(n =4), we have a real space with three dimensions
and time dimension, the equation is given by:
(34)
where is the potential energy of the three-dimensional object, is the inner product of
the potential energy in the real three-dimensional space, v is the velocity of the object, and is
the total energy of the object in the four-dimensional spacetime.
Hence, (35)
according to (31) (36)
using (20)
using (16)
, hence
(37)
so (38)
using (35) and (38)
so
hence (39)
using (31)
using (20)
using (16)
hence
(40)
The equation (40) like the equations (18) and (29) describes the total energy based on the rest
mass. At the speed of light where , we can simplify equation (40) to:
hence
(41)
using (37) (42)
hence (43)
Generalized Minkowski spacetime
In the Generalized Minkowski spacetime or n-dimensional Minkowski , we have a (n-1)-
dimensional real space and time dimension, the equation is given by:
(44)
Where is the potential energy of the (n-1)-dimensional object, is the inner product
of the potential energy in the (n-1)-dimensional space, v is the velocity of the object, and is the
total energy of the object in the n-dimensional spacetime.
Hence, (45)
using (18), (29), and (40) (46)
or
We define for simplifying the equations:
(47)
so (48)
According to (36), and (41) (49)
using (16) (50)
The equation (50) relates the potential energy of the (n-1) dimensional object in the n dimensional
spacetime to its rest mass.
or
hence (51)
(52)
using (49) (53)
using (16) (54)
, hence
(55)
in the higher speed where : (56)
At the speed of light where we can change equation (54) to:
so (57)
also, (58)
using (52) at the speed of light (59)
(60)
hence, (61)
using (57) and (61) (62)
Converting n and using n dimensional object instead of the n dimensional spacetime.
As mentioned before, in the Minkowski spacetime, n is the sum of the number of the real
dimensions and time dimension. On the other hand, in the current physics, physicists use the n as
the number of the real space dimensions. Hence, by converting n to n+1 in the equations we can
obtain current equations that physicists are using.
The equation (49) convert to: (63)
The equation (50) convert to:
(64)
The equation (53) convert to:
(65)
The equation (54) convert to:
(66)
The equation (55) convert to:
(67)
or (68)
where n is the number of the real dimension, is the potential energy of the n dimensional
particle , is the total energy of the particle, and , while .
Multidimensional energy-momentum equation
In the past century, physicists have used relativistic equation to relate the rest of the mass, the
momentum, and the total energy of the objects. This equation has been called the energy-
momentum equation: (69)
(70)
where
On the other hand, in this paper by using equation (69) where n is equal to 3, we can obtain new
energy-momentum equation. (71)
new energy-momentum equation is given by:
(72)
comparing equations (70) and (72) shows that at the right side of the equations there is some
differences. In the equation (70) or (69) which is known as the energy-momentum equation, the
power of the c is equal to 2, while at the equation (72) that is extracted from the multi-dimensional
energy equation, the power of the c is equal to 4.
Conclusion
By using the Generalized Minkowski spacetime, we obtained a new multi-dimensional energy
equation () where . By using approximation of the multi-
dimensional energy equation ( we obtained new energy momentum
equation () for three dimensional particles.
We presented a new equation that relates potential energy to the rest mass () where
n is the number of the real dimensions. Also , we have proved that the total energy of the particle
at the speed of light is almost equal to the c times of the rest potential energy () . At the
speed of light, the whole kinetic energy will be converted to the potential energy and we would
have new potential energy ( , hence the potential energy of the particle in the higher
dimension is the c times of the its potential energy in the lower dimension ( ).
Mass-energy equivalence equation in different dimension is different. For the one-dimensional
particles, . For the two-dimensional particles, , and for the n-dimensional
particles, . Hence, these equations show that Einstein mass-energy equivalence is
valid in the three-dimensional object where n=3.
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