Content uploaded by Kuan Peng
Author content
All content in this area was uploaded by Kuan Peng on Jul 15, 2021
Content may be subject to copyright.
1
Relativistic dynamics: force, mass, kinetic energy,
gravitation and dark matter
Kuan Peng 彭宽 titang78@gmail.com
14 July 2021
Abstract: Special relativity does not deal with acceleration, general relativity does not
deal with non gravitational acceleration, which leave the theory of relativity imperfect.
We will demonstrate some relativistic dynamical laws that specify relativistic
acceleration, force and kinetic energy. Also, based on equivalence principle does
gravitational mass vary with inertial mass?
1. Introduction ............................................................................................................................................ 1
2. Differential momentum equality ............................................................................................................. 2
a. Differential momentum ...................................................................................................................... 2
b. Proper inertial frame .......................................................................................................................... 2
c. Equality of differential momentum ..................................................................................................... 2
3. Inertial force ........................................................................................................................................... 3
a. Newtonian inertial force ..................................................................................................................... 3
b. Relativistic inertial force .................................................................................................................... 3
c. Transformation of relativistic inertial force ........................................................................................ 4
4. Gravitational force ................................................................................................................................. 4
a. Transformation of gravitational force ................................................................................................. 4
b. Gravitational force on moving and fixed bodies ................................................................................. 4
c. Gravitational mass of a moving body ................................................................................................. 5
d. Gravitational magnetic force and rotation curves of disc galaxies ....................................................... 6
5. Inertial mass and kinetic energy ............................................................................................................. 6
a. Relativistic inertial mass .................................................................................................................... 6
b. Relativistic kinetic energy .................................................................................................................. 7
c. Relativity and virial theorem .............................................................................................................. 7
6. Discussion .............................................................................................................................................. 8
1. Introduction
Newtonian kinematics defines motions of objects with velocity and acceleration,
Newtonian dynamics defines force with acceleration and mass, which makes
Newtonian mechanics the most complete theory in physics. Special and general
relativity are extremely successful, but they lack the capability of dealing with
acceleration and force. For relativity mass increases to infinity when u=c, which
makes energy and momentum to become incorrectly infinity. Also, general
relativity is based on equivalence principle according to which inertial mass is
equivalent to gravitational mass. Then does gravitational mass increases when
inertial mass increases? So, relativity needs new laws to deal with acceleration
and force.
In previous studies of relativity [1][2][3][4][5], we have already correctly treated
acceleration, inertial mass, kinetic energy. Below we will demonstrate the laws
that describe them. For setting the demonstrations on a strong base, we begin
with rigorously proving the equality of differential momentum in 2 relatively
moving frames of reference.
Notation convention:
Bold letters: vectors
(i): equation index
M, m: masses
M0, m0: rest masses of M, m
α: acceleration
c: speed of light
F: Force
P: momentum
t: time
u: velocity of a frame
v: velocity of an object
Subscript:
1:
Stationary frame
2:
Mobile frame
g:
Gravitational
u:
Moving at velocity u
a, b:
Object a and b
2
Note: In this article the word “relativistic” means “in the theory of relativity”, not moving at extremely high
speed. For example, “relativistic mass” means that a mass increases with velocity but not that it moves near the
speed of light.
2. Differential momentum equality
a. Differential momentum
The momentum of an object equals the product of its mass m and its velocity u as defined in equation (1). If the
object gets an infinitesimal impulse, it gets an infinitesimal change of momentum, which we call differential
momentum. If we see the object in 2 relatively moving frames of reference, the value of its momentum is
different in each frame. But, for Newtonian mechanics a differential momentum has the same value in all frames.
In relativity, a change of velocity has different value in relatively moving frames. However, differential
momentum has the same value in such frames, which we call equality of differential momentum and have
explained in « Velocity, mass, momentum and energy of an accelerated object in relativity » [2].
For rigorously proving this equality, let us take 2 identical objects labeled a and b. The object b moves at the
velocity u with respect to a, the frame of reference of the object b is labeled frm. b, see Figure 1. If the object b
gets an infinitesimal impulse, it gets a differential momentum and a differential change of velocity labeled dub.
b. Proper inertial frame
Notice that in the frame frm. b the velocity of b is constantly zero. Then, how
can its change of velocity dub be nonzero? In fact, dub is with respect to an
inertial frame, not to frm. b. For defining dub we have to create a new type of
inertial frame that coincides with b. We name this type of frame “Proper
inertial frame”.
For example, the object b moves at the instant velocity ut at a given time t. At
this time we create the proper inertial frame of b labeled Ref. b which moves at
constant velocity that equals ut. The trajectory of Ref. b is a straight line while
that of b is a curve, see Figure 2. After the infinitesimal impulse b moves at the
instant velocity u’t and the change of velocity equals dub = u’t  ut with ut being
the velocity of the inertial frame Ref. b. In the same way the proper inertial
frame of the object a is labeled Ref. a.
Proper inertial frame is a new notion, so we give its definition below:
Definition 1: Proper inertial frame of an accelerated object
At a given time t an accelerated object moves at the instant velocity ut. The proper inertial frame of this object at this time is a frame that
moves at the constant velocity ut.
c. Equality of differential momentum
Back to our 2 identical objects a and b and let them interact dynamically with each
other. During an infinitesimal time the 2 objects acquire the differential momentums
dPa and dPb in their respective proper inertial frame. Suppose that a and b are alone
in space, then dPa comes from b and dPb comes from a.
In Ref. b Newtonian mechanics applies because the velocity of b is zero. The
infinitesimal change of velocity of b is dub, its differential momentum is dPb which
equals with m0 being its rest mass, see equation (2). In the same way, the
differential momentum of a in Ref. a is dPa and is expressed in (3).
The frames Ref. a and Ref. b are both inertial. According to the principle of relativity,
Ref. a is not privileged over Ref. b and vice versa. Also, because the objects a and b are identical, their
dynamical behavior are exactly symmetrical. That is, if someone travels with one object and looks at the
trajectories of the other, he has no way to tell if he is with a or with b. So, the change of velocity of one object
must be exactly opposing that of the other. Then, we have dua =

dub, see (4). We multiply (4) with m0 and
obtain (5), which gives (6) by using (2) and (3).
Figure 1
Figure 2
(1)
(2)
(3)
Interaction
(4)
(5)
Using (2) and (3)
(6)
Ref. b
b
Trajectory
of b
Trajectory
of Ref. b
a
b
frm. b
u
3
On the other hand, the differential momentum of the object b can be specified in the
proper inertial frame of the object a which is Ref. a. We label this special differential
momentum with dP’b. So, dP’b is in the same frame as dPa and because of momentum
conservation dPa and dP’b cancel out in the frame Ref. a, see (7). By combining (6) and
(7) we obtain (8) which means that the value of the differential momentum of b is the
same in Ref. b and Ref. a. This is the equality of differential momentum for the object b
and is proven by using 2 identical objects. Is this equality valid in general?
For the object b there is no constrain about the position and velocity or about the kind
of force of interaction in the frame Ref. a. That is, b can be at any position and moves
at any velocity in Ref. a. So, b is in fact an arbitrary object in Ref. a and reversly, Ref. a
an arbitrary inertial frame with respect to b. Since dPb is specified in the proper inertial frame of b and dP’b in
Ref. a, the equality dPb = dP’b means that the differential momentum of b has the same value whether specified
in its proper inertial frame or in another arbitrary inertial frame. That is, this equality is valid in general.
For general use, we will refer the frame Ref. a as Ref. 1 and the frame Ref. b as Ref. 2, dP’b as dP1 and dPb as dP2,
see equation (9). Using (9) in (8) we get the equality dP1 = dP2 in (10), with dP2 being the differential momentum
of an accelerated object specified in its proper inertial frame and dP1 that specified in any other inertial frame.
We call equation (10) equality of differential momentum and state the following law:
Law 1: Equality of differential momentum
An object moves at the velocity u in the inertial frame Ref. 1. The proper inertial frame of the object is the frame Ref. 2. When the object gets
an infinitesimal impulse, the resulting differential momentum of the object specified in Ref. 1 and Ref. 2 has the same value.
The equality of differential momentum can be demonstrated directly from differential momentum as below
without using rest mass and velocity:
During an infinitesimal interaction between the objects a and b which are identical, the behavior of the objects a and b are exactly
symmetrical. According to relativistic principle, the proper inertial frame of a is not privileged over that of b and vice versa. So, the
differential momentum that the object a gets and that the object b gets are exactly opposing, which is expressed in equation (6). After
that, the equality of differential momentum is derived from (7) to (10).
As differential momentum is more abstract than velocity, this demonstration is independent
from the other one and makes the equality of differential momentum stronger.
3. Inertial force
a. Newtonian inertial force
Inertial force is defined as the time derivative of momentum, see equation (11). For
Newtonian mechanics, the mass m of an object is constant and the time derivative of its
momentum is in (13) with u being its velocity and
its acceleration, see (12). Then, we get
the Newton’s formula for inertial force in (14).
But in relativity m is not constant, then how is inertial force defined?
b. Relativistic inertial force
For relativity, momentum is still the product of m and u whereas m is not constant.
Inertial force is still defined as the time derivative of momentum, see (1) and (11).
Now we develop completely the time derivative of the product mu in (15). Because
the mass m is not constant, if u ≠0 the term
is nonzero and the time derivative
of momentum does not equal , see (16). So, the Newton’s formula for inertial
force (14) is not valid for relativity in general.
However, in the proper inertial frame of an object its velocity is always zero and the
term
in (15) is zero and the inertial force of the object equals with m0
being its rest mass, see (17). So, in relativity the Newton’s formula for inertial force
(14) is valid in the proper inertial frame of the object. As this inertial force changes
its value across frames, we call it “relativistic inertial force”.
Law 2: Relativistic inertial force
Relativistic inertial force of an accelerated object is defined as
and equals in the proper inertial frame of the object with m0
being its rest mass and
its acceleration in this frame.
Momentum
conservation
(7)
Using (6) and (7)
(8)
(9)
Using (8) and (9)
(10)
(11)
(12)
m constant
(13)
(14)
Using (1) and (12)
(15)
(16)
Proper inertial frame
, using (15)
(17)
4
c. Transformation of relativistic inertial force
Since relativistic inertial force changes its value, we need a transformation for
relativistic inertial force to know its value in inertial frames other than the
proper inertial frame. Let Ref. 2 be the proper inertial frame of an object
which moves at the instant velocity u in another inertial frame Ref. 1. Let P1
be the momentum of the object in Ref. 1 and t1 the time in Ref. 1. The inertial
force of the object in Ref. 1 is F1 and equals the derivative of P1 with respect
to t1, see equation (18). In the same way, the force in Ref. 2 is F2 and
expressed in (19).
We transform the derivative
in (20) where
is the ratio of time between
Ref. 1 and Ref. 2, see (21). The expression of
was derived in « Velocity,
mass, momentum and energy of an accelerated object in relativity » [2] , see
the equation (8) of this article.
Using the equality of differential momentum (10) we replace dP1 with dP2 in
(20), then we apply (21) for
in (20) and obtain (22). Using (18) and (19) in
(22) we obtain (23) which expresses F1 in terms of F2. Equation (23) is the
transformation for relativistic inertial force between inertial frames.
Law 3: Transformation of relativistic inertial force
An accelerated object gets an infinitesimal impulse. The resulting relativistic inertial force of the object in the inertial frame Ref. 1 and the
proper inertial frame of the object Ref. 2 are F1 and F2 respectively. The transformation of the relativistic inertial forces between
Ref. 1 and Ref. 2 is
with u being the relative velocity between the 2 frames.
4. Gravitational force
a. Transformation of gravitational force
Gravitational force between 2 bodies is defined by Newton’s universal law of
gravitation. Let M be an attracting body and m an attracted body. The gravitational
force on m is Fg and is expressed in (24) with M0 and m0 being the rest masses of M
and m respectively, R the distance between M and m, er the unit vector pointing from
M to m.
Gravitational force is not inertial force but creates an inertial force on the object it
accelerates. Let Ref. 1 and Ref. 2 be the proper inertial frames of M and m
respectively. In the case where m moves at the velocity u with respect to M, Fg2 the
gravitational force on m in Ref. 2 creates the relativistic inertial force F2 which
equals exactly Fg2, see (25). The transformation of relativistic inertial force (23)
applied to Fg2 gives F1 which is the relativistic inertial force specified in Ref. 1, see
(26), which is in fact created by the gravitational force in Ref. 1 labeled Fg1, see (27).
Using (27) in (26) we get (28), which transforms the gravitational force Fg2 into Fg1.
So, gravitational force also changes its value across frames.
But Newton’s universal law of gravitation gives only one value. Which of Fg1
and Fg2 is the correct one?
b. Gravitational force on moving and fixed bodies
In fact, Newton’s universal law of gravitation treats all bodies as stationary
even they move in space, for example, the moon and the planets. But for
relativity the velocities of bodies matter and the relative velocity between
attracting and attracted bodies has to be taken into account for gravitation.
Let m move at the velocity u with respect to M and Ref. 2 be the proper inertial frame of m. The proper inertial
frame of M is Ref. 1. M and its gravitational field are stationary in Ref. 1, see Figure 3. As gravitational force
equals the time derivative of momentum, gravitational field is a constant flow of momentum. The attracted body
collects a quantity of momentum per unit time which equals the gravitational force on it.
Using (11) in Ref. 1
(18)
Using (11) in Ref. 2
(19)
(20)
(21)
Using (10) and (21) in (20)
(22)
Using (18) and (19) in (22)
(23)
(24)
(25)
Using (25) in (23)
(26)
(27)
Using (27) in (26)
(28)
Figure 3
M
Gravitational
field
Ref.1
u
m
Ref.2
C
5
The circle C in Figure 3 is a small region fixed in the frame Ref. 1. Suppose that m
moves across the region in the time dt1 within which m is inside this region and
collects the quantity of momentum dPg1. Suppose now that m is stationary inside the
region. In this case m collects the quantity of momentum dP’g within the same time
dt1. When the size of the circle C is reduced indefinitely, the positions of the moving
and stationary m become the same, dPg1, dP’g and dt1 become infinitesimal. Because
M delivers the same quantity of momentum during dt1 whether m moves or not, we
should have dPg1=dP’g, see (29). Dividing (29) with dt1 gives (30). The 2 sides of this
equation are the gravitational forces Fg1 on the moving m and Fg on the stationary m,
see (31). Fg1 and Fg are in Ref. 1 because dPg1 and dP’g are in Ref. 1. So, we obtain
Fg1 = Fg in (32), with Fg defined in (24).
The above demonstration is rather tedious. Here is a simpler demonstration. Let A
and B be 2 fixed points in the gravitational field of M, see Figure 4. The body m does
a round trip from A to B then back to A. The velocity on AB trip is uo and that on BA trip is
ui. Let the gravitational force on the AB trip be F(uo) and that on the BA trip F(ui).
The differential work done by a force F on a differential distance dx is dw, see equation (33).
On the AB trip and the BA trip the differential works done by the gravitational force are
defined by (34). The total work done over the round trip is defined by (35). As gravitational
field is conservative the total work done over the round trip is zero, see (36), which is true for
the velocities uo and ui of whatever value and F(uo) cancels F(ui) out, see (36). Thus,
F(uo) = F(ui) for whatever velocity, see (37). In other words, whether
m moves or not, the gravitational force on m has the same value in
Ref. 1.
As (37) is true for the straight line AB in any direction, F(uo) = F(ui)
is true for the body m moving in any direction. So, equation (37) and
the statement “gravitational force on a body is independent of its
velocity” is true in general.
Law 4: Gravitational force is independent of velocity
Let M be an attracting body, m an attracted body. The gravitational force on m is
independent of the velocity of m in the proper inertial frame of M.
Applying (32) in (28) gives (38) which expresses Fg2 the gravitational
force on m in Ref. 2.
Law 5: Gravitational force in the proper inertial frame of the attracted body
Let M be an attracting body, m an attracted body. m moves at the velocity u with
respect to M. Let M0 and m0 be the rest masses of M and m respectively. The
gravitational force on m specified in the proper inertial frame of m is
.
It is important to underline the significances of (32) and (28):
In (32) Fg1 is the force on the moving m, Fg is the force on the stationary m. So, (32) expresses the
equality of the forces on 2 different bodies.
In (28), Fg1 is specified in Ref. 1 and Fg2 specified in Ref. 2, they are the same force on the same body
but specified in 2 different frames. Fg1 and Fg2 are the transformation of each other.
c. Gravitational mass of a moving body
In the proper inertial frame of the moving m, Ref. 2, the force on m is Fg2 which
equals its rest mass m0 multiplied by its acceleration
because Newtonian
mechanics applies, see equation (39). Combining (38) and (39) gives (40), which
expresses
the acceleration
in terms of M0 without m0. By reducing the term
into Mu, see (41),
is expressed in (42) where Mu acts as a gravitational mass.
(29)
(30)
(31)
Using (31) in (30),
then (24)
(32)
Figure 4
(33)
(34)
(35)
True for whatever uo and ui
(36)
(37)
Applying (24) in (28)
(38)
In Ref. 2
(39)
Using (38) and (39)
(40)
M
Gravitational
field
Ref.1
A
B
6
Because Mu increases with the relative velocity, we call Mu relativistic
gravitational mass of M. Reversely, in the proper inertial frame of M the relativistic
gravitational mass of m is
, see (43).
Mu is bigger than M, which means that in the proper inertial frame of m the
gravitational mass of M increases with velocity. So, for 2 gravitational bodies, the
relativistic gravitational mass of the attracting body increases with velocity. This
increase is symmetrical between the 2 bodies. Indeed, the term
in (38) can
be associated either with M0 or with m0.
Law 6: Relativistic gravitational mass
In the gravitational field of an attracting body M, an attracted body m moves at the velocity u. In the
proper inertial frame of m the relativistic gravitational mass of the attracting body is
with
M0 being the rest mass of M.
d. Gravitational magnetic force and rotation curves of disc galaxies
Because Fg2 is bigger than the Newtonian gravitational force Fg, see equations (38) and (24), the gravitational
force between 2 relatively moving bodies is bigger than when the 2 bodies are stationary. The increased part of
the gravitational force is proportional to the relative velocity squared, see equation (44), which is sort of like
magnetic force. This is the gravitational magnetic force that we proposed in «Magnetism and dark matter» [5]. In
a disc galaxy the stars rotate at big velocity and the gravitational force the stars feel should be bigger than if the
galaxy were still. So, the gravitational magnetic force makes the disc of galaxies rotating faster than expected.
If we suppose that most mass of a disc galaxy is concentrated in its center and use Newtonian mechanics to
compute the rotation curve of a disc galaxy, we will find a big difference between the computed curve and the
observed one. But, it was shown in «How galaxies make their rotation curves flat and what about dark matter?»
[3] that the Newtonian gravitational force of a mass distributed in the shape of a disc makes the rotation curves
already quite flat, which is the case of a disc galaxy. Thus, the difference between the computed Newtonian flat
curves and the observed flat curves is in fact quite small and could be explained with the gravitational magnetic
force.
5. Inertial mass and kinetic energy
a. Relativistic inertial mass
In Newtonian mechanics the inertial mass of an object is the coefficient m in
equation (14). We can also write equation (14) for relativity, but in this case the
coefficient m varies with the velocity of the object. We call this varying m the
relativistic inertial mass of the object and derive its variation law below.
Suppose that an object b moves at the velocity u in an inertial frame labeled
Ref. 1 and gets an infinitesimal impulse. In its proper inertial frame Ref. 2 the
resulting differential momentum and velocity are dP2 and dv2 respectively.
Because Newtonian mechanics applies in Ref. 2 we write equation (45). The
differential velocity of b specified in Ref. 1 is dv1 which is transformed from dv2
using (46) the transformation for differential velocity derived in « Relativistic
kinematics » [1], see the equation (15) of this article.
Combining the equations (45) and (46) gives the expression of dP2 in (47).
According to (10), dP2 = dP1, with dP1 being the differential momentum of b
specified in Ref. 1. Then we replace dP2 with dP1 in (47) to obtain the expression
of dP1 in (48). By dividing (48) with the differential time dt1 we obtain (49) which
is written into (50) with F1 being the relativistic inertial force of b in Ref. 1 and α1
its acceleration in Ref. 1.
From (50) we extract the term
and label it as mu, see (51). mu is the
(41)
Using (41) in (40)
(42)
Reversely for M
(43)
(44)
In proper inertial frame
(45)
(46)
(47)
Using (10) in (47)
(48)
(49)
(50)
(51)
7
relativistic inertial mass of b moving at the instant velocity u. Equation (51) is in the same form as that given in
« Introduction to Special Relativity » by James H. Smith [5], except that the latter was derived using a complex
process of shock.
Equation (51) is the variation law of relativistic inertial mass. Notice that equation (51) transforms m0 into mu
which are the inertial masses of b in 2 relatively moving frames. So, equation (51) is in fact the transformation
for relativistic inertial mass the same way as the transformation for time in relativity, see (21). For relativistic
gravitational mass also, equation (43) is the transformation for relativistic gravitational mass because it has the
same form as equation (51).
Since the relativistic inertial and gravitational masses of an object are transformed the same way, they will
always have the same value for whatever velocity of the object. This result illustrates well the equivalence
principle which stipulates that inertial and gravitational mass are equivalent. Imagine a space rock flies by the
Earth at any velocity, if we measure its inertial mass with one method and its gravitational mass with another
method, we will always find the same value for the 2 masses.
b. Relativistic kinetic energy
Since we have the expression of relativistic inertial force in equation (50), we
can compute the relativistic kinetic energy of a moving object by integrating
(50). An accelerated object acquires kinetic energy as work is continuously
done by the inertial force. Equation (52) expresses the relativistic differential
work dw that the relativistic inertial force F1 does over the differential distance
dx, with F1 being given by (50).
We transform the term from (52) in (53). The proper inertial frame of the
accelerated object is Ref. 2, u is its velocity and v1 that of the object, so u
equals v1, see (54). Then, the term
in (53) is transformed in (55).
Combining (55), (53) and (52) we obtain the expression of dw in (56).
When the object is accelerated from velocity zero to velocity u, the total work
equals the integral of (56) over the velocity zero to u and this work is
completely stored as kinetic energy in the object. The integration of (56) gives
(57) which is the expression of the relativistic kinetic energy of an object with
rest mass m0 moving at the velocity u. This expression was already derived in
« Velocity, mass, momentum and energy of an accelerated object in relativity »
[2], but the equality of differential momentum (10) was not proven at the time.
The constant of integration for (57) was determined to satisfy the condition of
the lower limit: u=0, . But one constant should not satisfy 2 conditions
at once. However, the upper limit condition: u=c, which is the
energymass equivalence is satisfied nevertheless, see (58). This is an amazing
but consistent result which makes (57) reliable.
Law 7: Relativistic kinetic energy
The relativistic kinetic energy of an object with rest mass m0 and moving at the velocity u equals
c. Relativity and virial theorem
The Caltech professor Fritz Zwicky was first to notice that, “The galaxies (in the neighboring Coma Cluster of
galaxies) were moving too fast within the cluster for the amount of illuminated stuff”, wrote Marlene Gotz in her
Presentation «Dark matter»[7]. Fritz Zwicky used the virial theorem [9] to draw this conclusion. Marlene Gotz
continued: “Analyzing the motions of all kinds of clusters shows that they cannot be stable unless there is a large
amount of mass than visible”, which marks the birth of the notion “Dark matter”.
The virial theorem that Fritz Zwicky used was derived using Newtonian kinetic energy. However the correct
expression of kinetic energy is the relativistic one given in (57). Could a relativistic virial theorem explain the
excess of velocity in clusters of galaxies?
Using (50)
(52)
(53)
(54)
(55)
Using (52) and (55)
(56)
(57)
(58)
8
Let us compare the velocities given by Newtonian kinetic energy and
relativistic kinetic energy in the case where the velocity of an object decreases
from uo to uN. Decrease of velocity and kinetic energy happens when objects
leave a gravitational field. For example, Oumuamua[10] and the Pioneer
spacecrafts[11] leaving the solar system. The Newtonian kinetic energy is
expressed by (59) using which we compute the variation of velocity squared
and the resulting difference of Newtonian kinetic energy E in (60).
In (61) we compute the variation of
velocity squared using relativistic
kinetic energy (57). is so computed to
give the same difference of energy E.
Dividing (61) by (60) we obtain (63). Then,
we transform (63) into (64) which we
multiply with to obtain (65).
Because kinetic energy decreases, E<0,
is negative, see (66). Starting from the
same velocity u0 for the 2 computations,
we obtain (67), which shows that the
final velocity squared given by relativistic
kinetic energy is bigger than the final
velocity squared given by Newtonian
kinetic energy. So, if a galaxy is far from
the center of its cluster, the velocity given
by relativistic kinetic energy would be
bigger than that predicted by the Newtonian virial theorem. So, the result
of a relativistic virial theorem would be closer to the observation.
In fact, equation (67) was already derived in « Analytical equation for
SpaceTime geodesics and relativistic orbit equation » [4] which
explained the unexpected “boost in speed” or the bigger than expected
velocity of Oumuamua[10] and the Pioneer anomaly[11].
If relativistic kinetic energy could explain the excess of velocity in
clusters of galaxies, could the clusters be held stable by gravitational force?
In the previous chapter we have shown that relativistic gravitational force
is bigger than Newtonian gravitational force, see (38). In a cluster of
galaxies, the gravitational attraction on one galaxy is acted by all the other
galaxies and increased by the big relative velocities with respect to each
one. So, the relativistic gravitational attraction could hold a cluster of
galaxies stable.
In consequence, relativistic kinetic energy as well as the increase of gravitational force could explain at least part
of the problem of fast moving galaxies in clusters. In addition, it's worth deriving the relativistic virial theorem
for space sciences.
6. Discussion
We have introduced the new notion “Proper inertial frame of an accelerated object” that allows defining
acceleration for relativity while special relativity cannot. Using this new frame of reference, the equality of
differential momentum is rigorously proven, new relativistic dynamical laws such as transformations for inertial
and gravitational forces and transformations for inertial and gravitational masses have been demonstrated. Also,
we have derived the relativistic expression of kinetic energy which satisfies the conditions of both the lower and
upper limits, which strengthens the consistency of this new expression.
We have derived relativistic inertial mass directly from the new relativistic dynamics, which shows the true
nature of the increase of relativistic inertial mass and gives a better understanding of special relativity. Also, we
have demonstrated that gravitational mass increases with velocity just like inertial mass does, which satisfies the
(59)
Newtonian E, using (59)
(60)
Relativistic E, using (57)
(61)
(62)
Dividing (61) by (60)
(63)
(64)
Using (64)
(65)
Using (65) and (62), E<0
(66)
Using (62) in (66)
(67)
9
equivalence principle. It turns out that the variation of mass with velocity is the transformation of mass between
2 relatively moving frames like the transformation of time.
We find that our relativistic dynamical laws could in principle explain part of the fast rotation of disc galaxies
and fast moving galaxies in cluster of galaxies. In fact relativistic dynamical effects increase the velocity of
space objects without emitting any electromagnetic wave, which is exactly the property of Dark matter. So, the
effect of the invisible “Dark matter” could be in reality relativistic effect. Of cause, this hypothesis needs to be
verified through further theoretical and experimental researches.
References
[1] « Relativistic kinematics » https://www.academia.edu/44582027/Relativistic_kinematics,
https://pengkuanonphysics.blogspot.com/2020/11/relativistickinematics.html
[2] « Velocity, mass, momentum and energy of an accelerated object in relativity »
https://www.academia.edu/42616126/Velocity_mass_momentum_and_energy_of_an_accelerated_object_in_relati
vity, https://pengkuanonphysics.blogspot.com/2020/04/velocitymassmomentumandenergyof.html
[3] «How galaxies make their rotation curves flat and what about dark matter?»
https://www.academia.edu/46903516/How_galaxies_make_their_rotation_curves_flat_and_what_about
_dark_matter https://pengkuanonphysics.blogspot.com/2021/04/howgalaxiesmaketheirrotation
curves.html
[4] « Analytical equation for SpaceTime geodesics and relativistic orbit equation »
https://www.academia.edu/44540764/Analytical_orbit_equation_for_relativistic_gravity_without_using
_Space_Time_geodesics https://pengkuanonphysics.blogspot.com/2020/11/analyticalorbitequation
for.html
[5] «Magnetism and dark matter» https://www.academia.edu/10183169/Magnetism_and_dark_matter,
https://pengkuanem.blogspot.com/2015/01/magnetismanddarkmatter.html
[6] « Introduction to Special Relativity » by James H. Smith
[7] «Dark matter» Marlene Gotz (Jena) https://www.hausderastronomie.de/3440769/09darkmatter.pdf
[8] https://en.wikipedia.org/wiki/Equivalence_principle
[9] https://en.wikipedia.org/wiki/Virial_theorem
[10] https://en.wikipedia.org/wiki/%CA%BBOumuamua
[11] https://en.wikipedia.org/wiki/Pioneer_anomaly