International Frontier Science
Letters, Vol. 15, pp. 9-14 (2020). https://doi.org/10.18052/www.scipress.com/IFSL.15.9
On the dependence of the relativistic angular momentum of a uniform ball
on the radius and angular velocity of rotation
Sergey G.
Fedosin
PO box 614088, Sviazeva str. 22-79, Perm, Perm Krai, Russia
E-mail:
fedosin@hotmail.com
In the framework of the special theory of relativity, elementary
formulas are used to derive the formula for determining the relativistic
angular momentum of a rotating ideal uniform ball. The moment of inertia of
such a ball turns out to be a nonlinear function of the angular velocity of
rotation. Application of this formula to the neutron star PSR J1614-2230 shows
that due to relativistic corrections the angular momentum of the star increases
tenfold as compared to the nonrelativistic formula. For the proton and neutron
star PSR J1748-2446ad the velocities of their surface’s motion are calculated,
which reach the values of the order of 30% and 19% of the speed of light,
respectively. Using the formula for the relativistic angular momentum of a
uniform ball, it is easy to obtain the formula for the angular momentum of a
thin spherical shell depending on its thickness, radius, mass density, and
angular velocity of rotation. As a result, considering a spherical body
consisting of a set of such shells it becomes possible to accurately determine
its angular momentum as the sum of the angular momenta of all the body’s
shells. Two expressions are provided for the maximum possible angular momentum
of the ball based on the rotation of the ball’s surface at the speed of light
and based on the condition of integrity of the gravitationally bound body at
the balance of the gravitational and centripetal forces. Comparison with the
results of the general theory of relativity shows the difference in angular
momentum of the order of 25% for an extremal Kerr black hole.
Keywords: relativistic angular momentum; moment of inertia; neutron star.
In the general theory of relativity,
the angular momentum of a ball can be calculated using the metric [1], since it
is necessary to take into account both the effect of gravity and centripetal
acceleration, which change the metric properties of the volume of the ball. In the special theory of relativity in the slow-rotation approximation,
the influence of the metric in the first approximation can be neglected.
The angular momentum of an ideal
uniform ball in classical mechanics is calculated by the formula:
,
(1)
where 
is the mass density of the ball’s
matter, 
is the angular velocity of rotation, 
is the ball’s radius, 
is the ball’s mass.
However, formula (1) does not take
into account the relativistic effect of the momentum’s dependence on the
velocity for each of the ball’s particles, so that (1) is applicable only at
low rotation velocities. According to the special theory of relativity, in a
body moving at a constant linear velocity, the mass density increases in
proportion to the Lorentz factor 
, where 
is the velocity of the body, 
is the speed of light.
At the same time, the volume of the
body decreases in inverse proportion to the Lorentz factor, as a consequence of
the Lorentz length contraction. As a result, the body mass as the product of the mass density
by the volume remains unchanged. The same applies to the mass element 
.
In the case under consideration, the
ball’s mass element is moving along the rotation circumference and not along a
straight line. Thus, our calculations will be limited to the accuracy, with
which the special theory of relativity approximates rotation of bodies, and
inertial reference frames approximate rotating non-inertial reference frames,
where acceleration of rotation occurs.
We will first try to calculate the
relativistic angular momentum of the ball in the Cartesian coordinates 
. In these coordinates, the volume element of a fixed uniform ball is
given by the formula 
. Multiplying the volume element by the mass density we find the mass element of the
ball: 
. Suppose the ball rotate at the
angular velocity 
about the axis 
of the fixed coordinate system with
the origin at the center of the ball.
Let some mass element be at a
distance 
from the ball’s center and have the
relativistic momentum 
, while the linear velocity of
motion of the mass element 
is given by the vector product of
the angular velocity 
by the radius-vector 
.
Since the vector is directed along the axis , only two components of the linear velocity are not equal to zero: 
. To calculate the angular momentum of
the mass element we need to multiply vectorially the radius-vector by the
momentum: 
. Then we need to integrate 
over all the ball’s mass elements
in order to find the total angular momentum 
. The components 
and 
are proportional to 
and 
, accordingly, so that after integration over the entire volume of the
ball, the components 
and 
will be equal to zero.
Since the vector has only one non-zero component 
which is directed along the axis , then taking into account the dependence of the Lorentz factor 
on the coordinates, for this
vector’s magnitude 
we find the following:
. (2)
We will substitute the limits of
integration in the volume integral (2) and integrate with respect to the
variable 
:
(3)
Apparently, the integral with
respect to the variable 
in (3) refers to elliptic
integrals and does not reduce to elementary functions, which makes it difficult
to calculate it. Therefore, the clearness is lost in predicting the dependence
of the angular momentum 
on the angular velocity, mass or radius of the ball.
Nevertheless, calculation of the
relativistic angular momentum of the ball is possible with the help of
elementary functions. We will illustrate this in the next section.
2. Calculation of the relativistic
angular momentum of the ball
In the cylindrical coordinate system 
, the volume element of a fixed uniform ball is defined by the formula 
, and the mass element of the ball is: 
.
The current coordinate 
is directed perpendicularly both to
the rotation axis and to the velocity . For the relativistic angular momentum of the ball’s mass element we
can write: 
. Let us cut the ball perpendicularly to the axis to a number of parallel layers
with thickness 
and calculate the angular momentum 
for one of such layers, which has
a certain maximum radius 
. Given that 
, 
, we have the following:
(4)
Now we need to integrate (4) over all the ball’s layers, that is, with
respect to the variable . If the ball’s radius is equal to , we can integrate with respect to the variable from zero to , that is, over one hemisphere, and then double the
result in order to take into account the second hemisphere. Inside the upper
hemisphere the radius of an arbitrary
layer is connected with the variable by the following relation: 
. This relation can be substituted into (4) and then
integrated over all the layers:
.
The result is as follows:
. (5)
At low angular velocities of
rotation , we can expand the logarithm in (5) up to the terms containing the
multiplier 
in the denominator. This gives the
standard angular momentum of the ball and the first order addition:
. (6)
Another limiting case is obtained if
we assume that the surface at the ball’s equator moves due to rotation at the
velocity reaching the speed of light. In (5) this corresponds to the fact that
the ratio 
tends to unity. If we take into
account that
,
then in (5) the third term with
logarithm disappears, and for the ball’s limiting angular momentum with 
we obtain:
. (7)
Within the framework of the special
theory of relativity no spherical body can reach the angular momentum equal to
(7). For gravitationally bound bodies there is a softer condition for the
maximum rotation velocity associated with the stability of matter at the
equator, where the linear velocity has the largest value. Here the centrifugal
acceleration must not exceed the acceleration from the gravitation force, which
leads to the inequality: 
. Substituting this angular velocity in (5) and in (6), we can estimate
the maximum angular momentum of gravitationally bound bodies, knowing only
their mass and radius.
In the general theory of relativity,
the extremal Kerr black hole with the largest possible rotation has the angular
momentum 
and the surface radius of events 
. If we substitute 
in (7), we will obtain 
. Thus, in the general theory of relativity the limiting angular
momentum of extremal objects increases by 25% in comparison with the result of
the special theory of relativity.
3. Calculation of the angular momentum of a neutron star and a proton
The results obtained are useful for
estimating the angular momentum and the moment of inertia of such rapidly
rotating objects as neutron stars and nucleons. The moment of inertia can be
determined as the ratio of the angular momentum to the angular velocity of
rotation: 
. The mass of a uniform ball depends
on the mass density and the volume of the ball: 
. Substituting it into (5), we obtain the formula for the moment of
inertia of a uniform ball rotating at the angular velocity 
:
. (8)
We will use (8) to estimate the
moment of inertia of the pulsar PSR J1614-2230, for which, according to [2], we
know the angular velocity of rotation
rad/s, the radius 
km, and the mass 
, where 
is the mass of the Sun. Since for this pulsar 
, the surface of its equator moves at velocity of about 8.5% of the
speed of light. With these data, it follows from (8)
that the pulsar’s moment of inertia is equal to 
kg×m2, and the angular momentum reaches the
value 
kg×m2/s. At the same time, if we would make
calculation using the classical formula 
, then the angular momentum would be equal to 
kg×m2/s. As we can see, in this case the
relativistic formula for the angular momentum gives ten times larger value than
the simple formula from classical mechanics. For a star, however, it should be
taken into account that its mass density increases at the center, where it
exceeds the average density approximately 1.5 times [3]. As a result, the
angular momentum of the star must be less than the value 
kg×m2/s, calculated for a
uniform ball in case of its relativistic rotation.
In quantum mechanics, it is known
that the proton’s spin is equal to 
, where 
is the Dirac constant. Suppose that this value is equal to the proton’s angular momentum 
, and let us estimate how the proton rotates in this case. If we denote 
, where 
is the proton radius, then from (8)
it follows:
. (9)
In (9) the Dirac constant 
and the proton mass 
are well known, and for the proton
radius we will use the value 
m, according to [4]. In this case (9) becomes the equation for 
. The solution of this equation is the
value 
, from which we obtain the value of the proton’s angular velocity of
rotation 
rad/s. In this case the velocity of
rotation of the proton’s surface reaches 30% of the speed of light.
4. Conclusion
Using division of a ball into a
number of parallel layers, we find in (5) a formula for the relativistic
angular momentum of a uniform ball expressed in terms of elementary functions.
According to (6), at low angular velocities of rotation the relativistic correction to
the standard angular momentum increases in proportion to 
, that is, in proportion to the cube of the rotation velocity. At sufficiently high rotation velocities the angular momentum changes as
a logarithmic function. The limiting angular momentum of the ball is reached
when the surface points at the equator of the ball move at the speed of light,
while the angular momentum is expressed by formula (7).
Formula (5) can be used to calculate
the relativistic angular momentum of a thin spherical shell with thickness 
: 
, where 
is the angular momentum of a ball
with the radius 
. In this case, a ball with reduced radius 
and angular momentum 
is embedded in this ball. Thus,
the angular momentum 
of the shell becomes a function
of the radius, angular velocity of rotation, mass density and thickness of this
shell.
With this in mind, any rotating
spherical body can be divided into a number of spherical shells, each of which
has its own radius, angular velocity of rotation and mass density, and,
accordingly, its own angular momentum . To calculate the angular momentum of a spherical body it is only
necessary to sum up the angular momenta of all the body’s shells. The accuracy
of the result will depend on the number of the shells used and on the accuracy
of the mass density distribution and of the angular velocity of rotation inside
the body.
We use the results obtained to
calculate the angular momentum and the moment of inertia of the neutron star
PSR J1614-2230. It turns out that the relativistic angular momentum is ten
times larger than the angular momentum according to the nonrelativistic
formula.
For a proton we determine the
corresponding angular velocity of rotation based on its quantum spin. In this case the velocity of the proton’s equatorial points reaches 30%
of the speed of light. As for the star PSR J1614-2230, the velocity of its
equatorial points reaches 8.5% of the speed of light. We need to add that at present the fastest rotating pulsar [5] is PSR J1748-2446ad with the angular velocity of rotation 
rad/s. If we assume that by mass and size it is the
analogue of PSR J1614-2230, then at the assumed radius km the relative velocity at the
star’s surface could reach the value 
. In this case, the star could rotate at the velocity at the equator of
the order of 19% of the speed of light, which is comparable to the rotation of
the proton surface.
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S.G. Estimation of the physical parameters of
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S.G. The radius of the proton in the
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