On the dependence of the relativistic angular momentum of a uniform ball on the radius and angular velocity of rotation

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.


Introduction
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: 5 2 0 where 0 ρ is the mass density of the ball's matter, ω is the angular velocity of rotation, a is the ball's radius, m 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 2  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 , , x y z . In these coordinates, the volume element of a fixed uniform ball is given by the formula dV dxdydz = . Multiplying the volume element by the mass density 0 ρ we find the mass element of the ball: Suppose the ball rotate at the angular velocity ω about the axis OZ of the fixed coordinate system with the origin at the center of the ball.
Let some mass element be at a distance × v ω R is given by the vector product of the angular velocity ω by the radius-vector R .
Since the vector ω is directed along the axis OZ , 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 dJ over all the ball's mass elements in order to find the total angular momentum J . The components x dJ and y dJ are proportional to x and y , accordingly, so that after integration over the entire volume of the ball, the components x J and y J will be equal to zero.
Since the vector J has only one non-zero component z J which is directed along the axis OZ , then taking into account the dependence of the Lorentz factor = we find the following: We will substitute the limits of integration in the volume integral (2) and integrate with respect to the variable z : Apparently, the integral with respect to the variable y in (3) refers to elliptic integrals and does not reduce to elementary functions, which makes it difficult to calculate it. Therefore, the clearness 10 IFSL Volume 15 is lost in predicting the dependence of the angular momentum J 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.

Calculation of the Relativistic Angular Momentum of the Ball
In the cylindrical coordinate system , , r z ϕ , the volume element of a fixed uniform ball is defined by the formula dV rdrd dz ϕ = , and the mass element of the ball is: The current coordinate r is directed perpendicularly both to the rotation axis and to the velocity v . For the relativistic angular momentum of the ball's mass element we can write: Let us cut the ball perpendicularly to the axis OZ to a number of parallel layers with thickness dz and calculate the angular momentum i dJ for one of such layers, which has a certain maximum radius Now we need to integrate (4) over all the ball's layers, that is, with respect to the variable z . If the ball's radius is equal to a , we can integrate with respect to the variable z from zero to a , 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 i r of an arbitrary layer is connected with the variable z by the following relation: 2 − . This relation can be substituted into (4) and then integrated over all the layers: The result is as follows: ω ω ω π ρ π ρ π ρ ω ω ω ω At low angular velocities of rotation ω , we can expand the logarithm in (5) up to the terms containing the multiplier 7 c in the denominator. This gives the standard angular momentum of the ball and the first order addition:

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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 a c ω tends to unity. If we take into account that 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: 3 Gm a ω ≤ . 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

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: J I ω = . The mass of a uniform ball depends on the mass density and the volume of the ball:

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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 3  , then the angular momentum would be equal to 41 5.11 10 × kg⋅m 2 /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 42 5.08 10 J = × kg⋅m 2 /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 2  , where  is the Dirac Thus, the angular momentum d J 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 d J . 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 3 4, 498 10 ω = × rad/s. If we assume that by mass and size it is the analogue of PSR J1614-2230, then at the assumed radius 12.8 a = km the relative velocity at the star's surface could reach the value 0.19 a c ω ≈ . 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.