This paper considers a model for the spherically symmetric space generated by a volume source of gravity having the shape of a ball of radius r₁ placed in a gravitational medium, which represents dark energy uniformly filling the entire space [1]. If the gravitational source were a point source, then the metric on the space would inevitably have a discontinuity at some finite distance from the source (which essentially means the presence of a black hole) and a discontinuity at the point where the gravitational source is located. However, in the model with a volume source of gravity, which is physically more realistic, the metric has no discontinuities under certain conditions on the size of the source and on the mass densities of the source and dark energy.

The question of how particles move in such a space inside and outside the volume source of gravity naturally arises. The accepted idea of the motion of point bodies in the case of a point gravitational source is described in [2, 3]. It is assumed that, in a certain neighborhood of the point source, bodies are attracted by the source according to Newton’s law, while outside this neighborhood (at sufficiently large distances), the presence of dark energy causes repulsion from the source according to Hubble’s law; thus, the radial velocity of particles is directly proportional to their distance from the source. Such a picture is typical of radial geodesics with respect to the absolute time s. These are precisely the geodesics subject to constraints with respect to the radial parameter r, which leads to the violation of the principle that a material body cannot travel faster than light. But if geodesics with respect to the world time t [4] are considered, then, even in the simplest case of a point gravitational source, the motion along radial geodesics obey more complicated laws, without the violation of the principle prohibiting motion faster than light.

The purpose of this paper is to describe properties of radial geodesics in the case of a volume source of gravity under various constraints on the parameters of the physical model.

Suppose given a pseudo-Riemannian spherically symmetric 4-space with signature and a ball of radius with constant mass density at the center of symmetry of this space. The entire space, including the ball, is uniformly filled with dark energy of constant mass density. The material ball (which may be a star or some other astronomical object) and dark energy determine the geometric properties of the space. Mathematically, the geometric properties of any pseudo-Riemannian space are completely determined by its metric. The interrelation between the mass distribution density in the space and the metric tensor, is described by the Equation (5)

where is the Ricci tensor, which is defined by

is the Christoffel symbol defined by

is the scalar curvature defined by

is the gravitational constant; and is the speed of light. Throughout the paper, summation over repeated indices is implied. Equation (1) directly implies the following relation between the scalar curvature of the pseudo-Riemannian space and the distribution density of the matter mass:

According to relation (5), the specification of the distribution density of the matter mass in the space is equivalent to the specification of the scalar curvature field of the pseudo-Riemannian space. The mass distribution density is governed by the gravitational field of the interaction of masses. According to the presently accepted point of view, the gravitational field is the metric on the pseudo-Riemannian space. Thus, the basic physical notions of gravitational theory, such as the density of the matter mass and the gravitational field, are interpreted in the language of differential geometry as the scalar curvature (up to proportionality) and the metric on the pseudo-Riemannian space, respectively. They are related by

this is a direct consequence of (1). Note that the system of Equations (5) and (6) is equivalent to (1).

A static spherically symmetric metric can be represented in spherical coordinates as

where, and, and are unknown positive functions. It follows from general geometric properties of a spherically symmetric space that the function must satisfy the condition

According to relations (2)-(5), the system (6) of gravity equations with respect to the components of metric (7) can be represented in the form

where and is the distribution density of the matter mass, which generally depends on the radial parameter. In [5], it was shown that if is a piecewise constant function having at most countably many discontinuities with respect to the parameter and is the mass of a ball of radius, then the spherically symmetric metric satisfying the system of Equations (9) has the form

where. The system of units used for measuring physical quantities is chosen so that. Metric (10) is a generalization of the Schwarzschild metric to the case of an arbitrary spherically symmetric mass distribution in a space with piecewise constant density. Note that metric (10) satisfies system (9) only on domains of continuity of the density.

We seek equations for geodesics by using the Lagrangian formalism. Let us introduce a Lagrangian of the form

where and.

Substituting Lagrangian (11) into the Euler-Lagrange equation

where x is one of the parameters, we obtain the following system of equations for the geodesics with respect to the parameter:

The system of differential Equation (12) can be integrated in the general form without any simplifying assumptions. As a result, we obtain

where and are integration constants; and must satisfy the condition

and and can be chosen arbitrarily. The physical meaning of is the angular momentum of the rotating body, and is the kinetic energy of the moving body. The functions, and are

We consider the radial geodesics in the equatorial planei.e., satisfying the conditions and. According to (12), we have; therefore, remains constant during the motion along the geodesic, and. The condition for a geodesic in the equatorial plane to be radial is

The radial geodesics with respect to the absolute time are given by the equations

For the model considered in this paper, we have

at and

at. Outside the ballthe acceleration and the velocity have the form

where. Inside the ball, the acceleration and the velocity have the form

Thus, outside the ball, on the interval

where, a test body experiences the action of an attractive force, while at, it experiences the action of a repulsive force. Inside the ball, a test body experiences the action of a braking force, and the acceleration is positive up to the center of the ball, at which it vanishes.

If, then the velocity outside the ball, which is determined by (18), vanishes at two points and. For the inequalities to hold, it is necessary that. Thusif

then and.

The motion of the particle begins at the rest position characterized by in the direction of the ball if the sign in (18) is “.” The acceleration of the test body on the interval is negative, i.e., the particle experiences an attractive force. As the particle passes through the surface of the ball, its velocity remains continuous, while the acceleration jumps by.

On the part of the trajectory inside the ball, the particle is subject to a braking force, and it stops at the position characterized by the radial parameter

Another motion from the rest state starts at the position characterized by the parameter. If the sign in (18) is “+,” then the motion of the particle starts at with positive acceleration, because. According to (17) and (18), the acceleration and the velocity of the particle unboundedly increase with, which contradicts the principle that a body cannot travel faster than light.

If, then the velocity of a test body does not vanish at any point outside the ball; inside the ball, it vanishes at the radial parameter

If, then (18) and (20) imply that the velocity of the particle nowhere vanishes, and (17) and (18) imply that as the radial parameter unboundedly increases, the velocity and the acceleration unboundedly increase as well.

We proceed to consider the radial geodesics with respect to the world time. In this case, we have

Taking into account (15) and (16), we obtain

where at. For the model considered in this paper, we have if and if; i.e.inside the ball, the space is described by the de Sitter metric [6], and outside the ball, it is described by the Schwarzschild-de Sitter metric. Formulas (22) and (23) for the acceleration and velocity outside the ball take the form

and inside the ball, they take the form

The radial geodesics can be divided into the two classes of trajectories determined by the conditions and. We conventionally refer to the trajectories from the first class as bounded and to those from the second class as unbounded [7]. In the next section, we consider these cases in more detail.

The analysis of properties of the radial geodesics with respect to the world time performed in this paper shows that these geodesics do not leave a certain spherically symmetric domain, provided that the radial motion starts at an interior point of this domain. This indirectly indicates that the presence of dark energy in a space renders this space finite and closed. Indeed, if the space was filled only with dark energy, i.e., if it had everywhere constant positive scalar curvature, then, as de Sitter showed in [6], this would be a finite closed elliptic space. Apparently, the finiteness and closedness of such a space must be preserved when a material body is placed inside it. However, the behavior of the radial geodesics with respect to the absolute time is alerting. Thus, the question on the finiteness and the closedness of the space in the model considered above remains open and requires additional analysis.