Compact stars: To cross or go around? That is the question

The travel times of light signals between two antipodal points on the surface of a compact star are calculated for two different trajectories: a straight line that passes through the center of the star and a semi-circular trajectory that connects the antipodal points along the surface of the star. Interestingly, it is explicitly proved that, for highly dense stars, the longer trajectory (the one that goes along the surface of the star) may be characterized by the {\it shorter} travel time as measured by asymptotic observers. In particular, for constant density stars we determine {\it analytically} the critical value of the dimensionless density-area parameter $\Lambda\equiv 4\pi R^2\rho$ that marks the boundary between situations in which a direct crossing of the star through its center has the shorter travel time and situations in which the semi-circular trajectory along the surface of the star is characterized by the shorter travel time as measured by asymptotic observers [here $\{R,\rho\}$ are respectively the radius of the star and its density].


I. INTRODUCTION
Many of us have encountered situations in which we have to drive from one side of a city to its opposite side.Then the dilemma arises: Should I choose the short route that passes through the center of the city, or should I choose the longer road that goes along the perimeter of the city?
The answer to this 'existential' question obviously depends on the density of traffic (the traffic congestion) that characterizes the city in question.In particular, one expects that there is a critical traffic load (critical car density) above which it would be better for us to choose the longer but less busy road, the one that connects one side of the city to its opposite side along the perimeter of the city.
Within the framework of general relativity, time as measured by asymptotic observers in curved spacetimes is influenced by the spatially-dependent energy density that characterizes the spacetime in question.This is one of the fundamental and most known predictions of Einstein's theory of general relativity.Suppose a physicist (Alice) has a remote controlled spaceship that can move arbitrarily close to the speed of light and she wants to send it from a point A on the surface of a compact star to its antipodal point B which is located on the opposite side of the star.And suppose Alice can choose between two options: (1) To send her remote controlled spaceship from the point A to the point B along a straight line that crosses the star directly through its center [1], or (2) To send the spaceship along a semi-circular trajectory that goes along the surface of the star from point A to its antipodal point B.
The following important question naturally arises: Which of these two journeys between the two opposite sides of the compact star will take less time as measured by Alice who is located far away from the compact star?
The answer to this physically interesting question is quite obvious in the flat-space limit of highly dilute stars, in which case the travel time is mainly determined by the length of the trajectory.In particular, the straight-line trajectory from a point A to its antipodal point B that crosses the star through its center is characterized by the shorter traveling time.
On the other hand, curved spacetime effects are expected to become important as the dimensionless density parameter [2] Λ ≡ 4πR 2 ρ max (1) that characterizes a star of radius R and maximum density ρ max deviates significantly from zero.In particular, for highly dense stars, the longer road which takes the remote controlled spaceship from the point A to its antipodal point B along a semi-circular trajectory on the surface of the star may have the shorter traveling time as measured by the remote operator (Alice).
The dimensionless density-area parameter Λ, which characterizes a star of radius R and maximum energy density ρ max , is a physically important quantity whose value can in principle be bounded from below by far away observers who measure the gravitational redshift factor of spectral lines that were emitted from the observed star [3].In the present compact paper we shall demonstrate that, due to this gravitational redshift (time dilation) effect in highly curved spacetimes, the identification of the trajectory that connects two antipodal points on the surface of a compact star and has the shorter travel time as measured by asymptotic observers (the remote operator) is a highly non-trivial task.
In particular, for some models of compact stars one may expect to find a critical value Λ = Λ * for the dimensionless density parameter that marks the boundary between situations in which a direct crossing through the center of the star has the shorter travel time as measured by asymptotic observers and situations in which the semi-circular trajectory along the surface of the star has the shorter travel time.
The main goal of the present compact paper is to demonstrate, using an analytically solvable model, the existence of a critical value for the dimensionless density-area parameter Λ of a compact star beyond which the longer orbit (the semi-circular trajectory along the surface of the star) has the shorter travel time as measured by the remote operator.

II. COMPACT STARS: TO CROSS OR TO GO AROUND?
We consider a spatially regular compact star of radius R whose asymptotically flat curved spacetime is described, using the Schwarzschild spacetime coordinates, by the spherically symmetric line element [4-7] where A = A(r) and B = B(r).
An asymptotically flat spacetime is characterized by the radial functional behaviors [6,7] A(r → ∞) → 1 and of the metric functions.In addition, a regular spacetime is characterized by the functional relations [6,7] A(r → 0) > 0 and at the center of the star.We shall assume that the energy density and pressure are zero outside the surface r = R of the compact star, in which case the spacetime outside the star is characterized by the Schwarzschild line element (2) with where M is the total (asymptotically measured) mass of the star.
Our goal is to minimize the crossing time T of the compact star by the remote controlled spaceship as measured by the remote operator (Alice).We shall therefore assume that, by using non-gravitational forces, the spaceship can move arbitrarily close to the speed of light along a non-geodesic trajectory.
In this case the crossing time T c of the star along a straight-line trajectory that connects a point A on the surface of the star to its antipodal point B and passes through the center of the star as measured by the asymptotic observers can be obtained from the curved line element (2) with the properties In particular, substituting Eq. ( 6) into (2) one obtains the integral relation for the travel time through the center of the star.
On the other hand, the travel time T s between the two antipodal points along a semicircular trajectory on the surface of the star as measured by the asymptotic observers can be obtained from the line element (2) of the curved spacetime with the properties ds = dr = dθ = 0 and ∆φ = 2π .
In particular, substituting Eqs. ( 5) and ( 8) into Eq.( 2) and performing the azimuthal integration, one obtains the functional expression for the traveling time along a semi-circular trajectory on the surface of the star.The travel time (9) can be expressed in the dimensionless form where C ≡ M/R is the characteristic compactness parameter of the star.
In order to facilitate a fully analytical treatment of the physical system, we shall illustrate our ideas using the analytically solvable model of constant density stars, which are characterized by the functional relations [5] and for the dimensionless metric functions inside the compact star.
The metric functions (11) and ( 12) can be expressed in a mathematically compact form in terms of the dimensionless density parameter Λ of the star [see Eq. ( 1)]: and where we have used here the dimensionless radial coordinate Substituting Eqs. ( 13), (14), and (15) into Eq.( 7), one finds the integral relation Interestingly, and most importantly for our analysis, one finds that the integral (16) can be evaluated analytically to yield the dimensionless expression for the travel time through the center of the star.
Taking cognizance of the analytically derived expressions (10) [8] and (17) for the travel times between the antipodal points of the compact star, one finds that the dimensionless ratio T c (Λ)/T s (Λ) is a monotonically increasing function of the dimensionless density parameter Λ of the star with the asymptotic properties and [9, 10] The asymptotic functional behaviors ( 18) and ( 19), which characterize the Λ-dependent dimensionless ratio T c (Λ)/T s (Λ), reveal the fact that there must exist a critical value Λ = Λ * of the dimensionless density-area parameter above which the longer trajectory that connects the antipodal points of the star (the semi-circular trajectory that goes along the surface of the star) is characterized by the shorter travel time as measured by the remote operator (Alice).
The critical density parameter of the constant density stars is defined by the functional relation or equivalently [see Eqs. ( 10) and ( 17)] This is a highly non-trivial equation for the critical value Λ * of the dimensionless density parameter that characterizes the family of constant density stars.
At first glance it seems that numerical methods must be used in order to solve Eq. ( 21).
Interestingly, however, despite the non-linear character of Eq. ( 21), the critical density parameter Λ * can be determined analytically if one notices the relations In particular, using Eq. ( 22) one finds from Eq. ( 21) that the critical density parameter, which marks the boundary between constant density stars that are characterized by the relation T c < T s to compact stars that are characterized by the opposite relation T s < T c , is given by the dimensionless value

III. PHYSICAL MODELS IN WHICH IT IS ALWAYS BETTER TO CROSS THROUGH THE CENTER OF THE COMPACT STAR
In the previous section we have explicitly proved that, for constant density stars, there is a critical density parameter [see Eqs. ( 1) and ( 23)] above which the longer trajectory (the one that goes along the surface of the compact star) is characterized by a travel time which is shorter than the crossing time through the center of the star.In the present section we shall explicitly prove that there are physical models of compact stars in which the radial trajectory that passes through the center of the star is always (that is, for all allowed values of the physical parameters that characterize the star) characterized by the shorter travel time.
To that end, we shall consider the physical model of thin-shell gravastars whose metric functions are characterized by the radial functional relations Substituting Eq. (24) into Eq.( 7) one obtains the integral relation The integral (25) can be evaluated analytically to yield the dimensionless expression for the travel time through the center of the compact star.
From the analytically derived expressions (10) with the asymptotic functional behaviors and [11] Interestingly, from Eqs. ( 27), (28), and (29) one deduces that, for all physically allowed values of the dimensionless compactness parameter C ∈ (0, 1/2) that characterizes the compact gravastars, the radial trajectory that connects the two antipodal points of the gravastar and passes through its center is characterized by the shorter travel time as measured by the remote operator (Alice).

IV. SUMMARY
The dimensionless density-area parameter Λ ≡ 4πR 2 ρ max of compact stars is an important physical quantity that in principle can be bounded from below, using the general relativistic gravitational redshift effect, by far away observers [3].In the present compact paper we have explicitly demonstrated that the value of this density parameter may determine the preferred route to be taken by a spaceship whose remote operator (Alice) wants to send it from point A, which is located on the surface of a compact star, to its antipodal point B.
Using the characteristic line element (2) of a compact star in a spherically symmetric curved spacetime, we have determined the travel times as measured by the remote operator for two different trajectories of the spaceship: a straight line that passes through the center of the star [1] and a semi-circular trajectory that goes along the surface of the star.Interestingly, we have explicitly proved that, similarly to what happens in cities with heavy traffic congestion, there is a critical density beyond which it is preferred to choose the longer path that connects the two antipodal points of the star along its perimeter.
In particular, using the analytically solvable model of constant density stars, we have revealed the existence of the critical value [see Eqs. for the dimensionless density parameter of the star above which the longer trajectory between the antipodal points (the one that goes along the surface of the star) is characterized by the shorter travel time as measured by the remote operator who is located far away from the star.
It is worth noting that the analytically derived critical density parameter (30) corresponds to a constant density star whose dimensionless compactness parameter (mass-to-radius ratio) C ≡ M/R has the critical value Finally, we have explicitly proved, using the analytically solvable model of thin-shell gravastars, that there are physical situations in which the radial trajectory that passes through the center of the star is always (that is, for all physically allowed values of the parameters that characterize the compact stars) characterized by the shorter travel time between the two antipodal points of the compact star.
and (26), which determine the compactness-dependent travel times between the two antipodal points of a star, one finds that, for the physical model of compact gravastars, the dimensionless ratio T c (C)/T s (C) is a monotonically decreasing function of the dimensionless compactness parameter C of the star, (1) and (23)] Λ ≡ 4πR 2 ρ = 1 (30)