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Universality of the Buchdahl sphere

Buchdahl sphere, the limiting stable isotropic stellar structure without exotic matter, plays a very important role, as far as the near horizon physics is concerned. This is because, the radius of the Buchdahl sphere lies between the horizon and the photon sphere, such that the signal arising from gravitational wave and black hole shadow will mimic that of a black hole. Here, we show certain universal properties associated with the Buchdahl sphere, in the sense that these properties will not change even if electric charge is included in the stellar structure. Moreover, these universal properties transcend general relativity and holds good in pure Lovelock theories of gravity as well. Using these universalities, we have proposed a Buchdahl limit for rotating stellar configuration. Finally, the universality of the Buchdahl limit in terms of the gravitational and non-gravitational field energies and also in terms of the photon sphere is also discussed.


Introduction
Recent gravitational wave measurements [1][2][3][4][5][6][7], as well as the observation of black hole shadow [8][9][10][11], have probed the characteristics of spacetime near the event horizon for the first time. Though the resultsby and large -are consistent with that of a general relativistic black hole, there is room for alternatives [12][13][14][15][16][17][18]. In particular, if stellar structures can exist within the region between the black hole horizon and the photon sphere, the gravitational wave signal, within the current experimental errors, will very much look like that of a general relativistic black hole [19][20][21][22][23][24][25][26]. Therefore, the existence of such stellar structures and study of their physical properties are always encouraging in its own right. Since, the existence of such stellar structures may provide alternatives to the black hole paradigm and thereby resolving the hitherto unsolved singularity problem.
In this connection, it is worthwhile to ask the limiting stellar structure possible, before it collapses to a black hole, purely from a gravitational point of view and the stability of the stellar structure. Such a limit was derived by Buchdahl long ago [27] and still is the limiting stellar configuration that is possible within the realm of gravitational physics, with generic stability arguments about the isotropic stellar material. This demonstrates the pivotal role played by the Buchdahl limit in this context. We have also noticed in [28][29][30][31][32][33][34][35][36][37][38] that depending on the matter content, spacetime dimensions and the nature of the gravitational action, the Buchdahl limit would indeed be different. However, this is still the limiting structure, which is possible for a stellar configuration, made out of non-exotic matter fields.
Here we demonstrate some new exciting properties of the Buchdahl sphere, which is a star with radius equal to the Buchdahl limit. First of all, we show that the escape velocity from the Buchdahl sphere is universal, i.e., the escape velocity remains the same irrespective of whether the stellar configuration is with or, without any electric charge. This then prompts one to assess the Buchdahl limit of a rotating stellar configuration for the first time. Note that, we do not know any solution of the Einstein's field equations, interior to a rotating stellar configuration. However through the universality of the escape velocity, it is indeed possible to obtain the Buchdahl bound on such rotating stellar configuration. Therefore, despite absence of any interior solution to a rotating geometry, we can argue that the radius of such stellar configuration must be less than a certain value, which we will derive in this work. To our surprise, this universality also translates into higher curvature theories of gravity, e.g., in the context of pure Lovelock theories. In addition to the escape velocity, the universal nature of the Buchdahl sphere also reflects in the expression for the gravitational potential and also in the partition between the quasi-local gravitational and non-gravitational energies. This further demonstrates the special role played by the Buchdahl sphere in various characteristic properties of the spacetime geometry. In particular, these universal properties of the Buchdahl sphere very much remind us of the properties of the black hole spacetimes [39], transcending general relativity. Thus it will be worthwhile to pursue further properties of the Buchdahl sphere,á la those of black hole spacetimes. Also note that the analysis presented here is for isotropic fluid sphere, thus following [40], it will be interesting to observe whether such universal properties will hold true for anisotropic Buchdahl sphere as well. These will not only have implications for stellar structures and its collapse to a black hole, but also on the ringdown signal from merger of binary compact objects and black hole shadow.
The paper is organized as follows, in Section 2 we show that the escape velocity, as well as the gravitational potential, associated with the Buchdahl sphere is indeed universal in four dimensional general relativity. Then in Section 3, we show that the universality of the Buchdahl sphere transcends general relativity and holds in pure Lovelock theories as well, with or without electric charge. Then we apply this universality in the context of rotating stellar structure with or without electric charge in Section 4. Finally, such universal properties of the Buchdahl sphere have been extended in the context of quasi-local gravitational energy in Section 5 and then we conclude with discussion on our results and possible future directions.
Notations and Conventions: Throughout this article we have followed the mostly positive signature convention, such that the flat spacetime metric in the Cartesian coordinate system takes the form, η µν = diag.(−1, +1, +1, +1). Furthermore, we have taken the fundamental constants, such that c = 1 = G.
2 Universality of escape velocity and potential in four-dimensional Einstein gravity In this section we will show that the escape velocity from a Buchdahl sphere in four-dimensional general relativity is universal irrespective of the matter content. For this purpose, we will first present the escape velocity from the surface of a static and spherically symmetric star. This requires one to define notion of the velocity of a particle in a coordinate invariant manner. However, the velocity must be observer dependent, for different observer will observe different velocities for the same particle. Following [41], one can write down the velocity of a particle with four-velocity u µ with respect to an observer with four-velocity u µ obs , which takes the form, For a static and spherically symmetric spacetime, the metric describing the spacetime geometry outside the stellar structure can be expressed as, g µν = diag.(−f (r), {1/f (r)}, r 2 , r 2 sin 2 θ). Since we are interested in the escape velocity, we take u µ to be the four velocity of an outgoing geodesic and u µ obs is taken to be a static observer at radius r, determining the velocity of the outgoing geodesic at radius r. Thus for the static observer at radius r, the four-velocity is given by, u obs µ = ( f (r), 0, 0, 0), while for the timelike outgoing geodesic, u 0 = {E/f (r)}, where E is the energy of the particle. Thus, we obtain u µ u obs µ = (E/ f (r)). Therefore, according to Eq. (1), the velocity of the outgoing geodesic with respect to a static observer is given by, The above expression can also be derived along the following lines -first of all from the line element of static and spherically symmetric spacetime we obtain, For motion on the equatorial plane, we can rewrite this expression as, For outgoing geodesic, it follows that,ṫ = {E/f (r)} and hence Eq. (2) immediately follows. In order to derive the escape velocity, it is necessary that the particle should just be able to reach infinity, i.e., the particle should be at rest at infinity. Thus setting v = 0, as f (r) → 1, in Eq. (2), we obtain E = 1. Thus the escape velocity of a particle from a radius r, in a static and spherically symmetric spacetime is given by, This is the expression for the escape velocity v esc we will use in order to derive its universal bound for Buchdahl sphere. In what follows we will use the Schwarzschild as well as the Reissner-Nordström spacetime in order to determine the escape velocity from the surface of a spherical star with or without electric charge.

Schwarzschild spacetime
Let us consider the spacetime outside a massive static and spherically symmetric star, which is given by the Schwarzschild solution. For which the unknown function f (r) appearing in the line element takes the form, f (r) = 1 − (2M/r) and hence from Eq. (3) the escape velocity of a massive particle from the surface of the star with radius R becomes, Note that the above escape velocity coincides with the corresponding Newtonian expression. If we now consider the star to be on the limit of stability, i.e., the radius of the star satisfying the Buchdahl limit, with R = (9/8)(2M ), it follows that, This provides the limiting escape velocity from the surface of a massive static and spherically symmetric star. Another interesting feature is the gravitational potential at the surface of the Buchdahl sphere. In general the gravitational potential Φ at the surface of a star with radius R is given by, Φ = −(M/R), such that on the Buchdahl sphere the gravitational potential becomes, Φ buchdahl = −(4/9). We will now demonstrate that the same limiting velocity and gravitational potential applies even in the presence of the electric charge.

Reissner-Nordström spacetime
Let us now consider the case of a charged, static and spherically symmetric shell with mass M and electric charge Q. The exterior geometry is given by the Reissner-Nordström spacetime, for which the unknown function f (r) takes the form, f (r) = 1 − (2M/r) + (Q 2 /r 2 ). In which case, the escape velocity from the surface of this spherical shell (with radius R) follows from Eq. (3) and is given by Note that for α = 0, the above escape velocity reduces to the corresponding expression for Schwarzschild spacetime. If we now consider the radius of this charged shell to coincide with the corresponding Buchdahl bound, which takes the form [30,31,37,38,42], then from Eq. (6) the limiting form of the escape velocity from its surface becomes, Therefore, we observe that whether the exterior geometry is Schwarzschild or, Reissner-Nordström, the escape velocity from the Buchdahl sphere is always the same, given by v 2 limit = (8/9). This suggests that the appearance of a limiting radius for stellar structure effectively originates from the existence of a universal escape velocity for four dimensional general relativity.
Note that for α 2 = (9/8), the Buchdahl sphere satisfies the condition, (M/R) = (8/9), which is twice of the (M/R) ratio for a Buchdahl sphere in vacuum. This is identical to the extremal black hole scenario, where the horizon radius of an extremal Reissner-Nordström black hole is precisely twice of the horizon radius for Schwarzschild black hole. Moreover, the above suggests that for 1 < (Q/M ) < (3/2 √ 2), there can be stable stellar structure which cannot collapse to a black hole, but possibly to a naked singularity. Thereby providing a formation channel for naked singular spacetimes. Finally, in the present scenario, the gravitational potential at the surface of a star with radius R reads, Φ = −(M/R) + (Q 2 /2R 2 ), such that on the Buchdahl sphere, this gravitational potential becomes, Φ buchdahl = −2v 2 limit = −(4/9), identical to that of the Schwarzschild black hole. This shows that both the escape velocity and the gravitational potential is universal for Buchdahl sphere with or without charge. We will now demonstrate that this result transcends general relativity and holds true for higher curvature theories as well, which will put forward the universality of the escape velocity from the Buchdahl sphere to a firm ground and then we will use the same subsequently, in obtaining the buchdahl bound for rotating star as well.
3 Universality of escape velocity and potential in pure Lovelock theory of gravity The above analysis motivates one to examine if such a universal nature of the escape velocity and the gravitational potential exists beyond general relativity as well. For this purpose we will discuss if such universal behaviour transcends general relativity. Following which, we will study the escape velocity from Buchdahl sphere and the gravitational potential on the surface of the Buchdahl sphere, with or, without the electric charge in d spacetime dimensions for pure Lovelock theory of order N . As in general relativity, we will first present the case of vacuum solutions before discussing the charged case. At this outset, let us also briefly describe the structure of the pure Lovelock theory. The Lovelock theories of gravity has a special place within the domain of higher curvature theories of gravity, since despite having higher curvature terms, these theories yield second order gravitational field equation, thereby exorcising the Ostrogradsky instability. In general, the Lovelock Lagrangian in d spacetime dimensions involve k ≤ (d/2) number of terms, each of which is a polynomial involving the Riemann tensor, such that the N th term in the series involves N Riemann tensors appropriately contracted with the alternative tensor. The pure Lovelock theories of gravity, on the other hand, corresponds to a single term in this series. For a pure Lovelock theory of order N , we have N Riemann tensors combined together with an alternative tensor of appropriate rank. Such a pure Lovelock theory of order N is dynamical in spacetime dimensions d ≥ 2N + 2. This is the scenario considered below.

Exterior vacuum spacetime
Let us consider the case of a static and spherically symmetric stellar configuration with vacuum exterior. Since the gravitational interaction is considered to be described by pure Lovelock gravity of order N in d spacetime dimensions, the unknown function f (r) appearing in the diagonal metric [43][44][45][46]. Note that for N = 1 and d = 4, the function f (r) becomes identical to that of the Schwarzschild spacetime. Therefore from Eq. (3), the escape velocity of a particle from the surface of the star with radius R, within the pure Lovelock theories of gravity reads, It is to be noted that, for N = 1 and d = 4, which is the four dimensional general relativity, the above expression for escape velocity reduces to the one presented in Eq. (4), as expected. The mass and radius of the Buchdahl sphere for the N th order Lovelock gravity in d spacetime dimensions, on the other hand, are related by the following relation [32], whose substitution in the expression for the escape velocity in Eq. (9) yields the following expression for the limiting velocity from the surface of the Buchdahl sphere, This limiting velocity is independent of the mass and hence is an universal feature, depending alone on the order of the Lovelock Lagrangian and the spacetime dimensions. Further, the gravitational potential on a sphere of radius R in pure Lovelock gravity reads, Note that for N = 1 and d = 4, the above expression for limiting velocity becomes (8/9), the universal velocity for four dimensional general relativity. Also the gravitational potential on the Buchdahl sphere reduces to −(4/9), as expected. Moreover, for d = 3N +1, we obtain, v 2 limit = (8/9) and Φ Buchdahl = −(4/9), which once again demonstrates the special role played by this spacetime dimension for pure Lovelock theory of order N , as the results are consistent with those of four dimensional general relativity (see [32,33,46] for a few more similar scenario).

Charged solution
To see if the above expression for limiting velocity as in Eq. (11) holds true in the presence of electric charge as well, as in the case of general relativity, here also we consider a static and spherically symmetric Buchdahl charged shell. For the charged exterior solution, the unknown function f (r) becomes, (3) the escape velocity from the surface of the charged shell with radius R takes the form, The Buchdahl sphere, on the other hand, satisfies the following condition between the mass M , radius R and electric charge Q of the spherical shell [38], Substitution of the same in the expression for escape velocity yields the following limiting expression for escape velocity from the Buchdahl sphere, which coincides identically with the corresponding one derived for the vacuum exterior. Therefore, given the order of the Lovelock polynomial and the spacetime dimensions, the escape velocity from the Buchdahl star is universal, i.e., independent of the presence of electric charge. Moreover, for any lovelock theory of gravity of order N , in the spacetime dimension d = 3N + 1, the limiting velocity reads v 2 limit = (8/9). Thus the existence of a universal escape velocity from a Buchdahl sphere transcends general relativity. The same limiting velocity holds true for pure Gauss-Bonnet gravity in seven dimensions and so on. Finally, the gravitational potential, which for the charged case is given by, -identical to that of the vacuum spacetime considered above. Thus the universality of the gravitational potential also transcends general relativity and holds in pure Lovelock theories of gravity as well. We will now use this universal escape velocity in order to determine the Buchdahl limit for rotating stars.

Buchdahl bound for rotating star
The Buchdahl bound for a rotating star, using the gravitational field equations in its interior, is very difficult to arrive at, since the field equations with matter in the presence of axi-symmetry is enormously complicated. Fortunately, the present analysis relating the existence of a limiting universal escape velocity with the Buchdahl sphere, suggests that the Buchdahl bound for rotating objects can be obtained through the determination of the escape velocity in such rotating spacetimes. In this section, we will first present a general expression for the escape velocity in a rotating spacetime, which we will use subsequently to find out the Buchdahl bound.
For axi-symmetry and stationary spacetimes, there are five metric functions that govern the spacetime geometry. We assume that these functions are such that the motion on the equatorial plane, located at θ = (π/2), is possible. For such a scenario, the trajectory of a particle on the equatorial plane in the rotating spacetime satisfies the following condition, where, e 2ν , ω, e 2ψ and e 2µ2 are the four unknown functions governing the geometry of the equatorial plane. In general these are all functions of the radial coordinate r and angular coordinates θ, however for the equatorial motion, these are functions of the radial coordinate alone. In order to obtain the escape velocity, we re-write the above expression as, from which the velocity of a massive particle particle moving on the equatorial plane takes the following form, However, this expression for the velocity of a massive particle on the equatorial plane is not so useful for our purpose. Rather, It is possible to derive an alternative expression for the velocity from Eq. (16), which can be expressed in terms of (dt/dτ ) as, The quantity (dt/dτ ) can be expressed in terms of the conserved energy E (the metric elements are independent of time t) and conserved angular momentum L (the metric elements are independent of angular coordinate φ), per unit mass, such that, Substitution of the above expression for (dt/dτ ) in the velocity expression as in Eq. (18) yields, In order to proceed further and to relate this velocity with that of escape velocity from the surface of rotating star, we need to impose the following boundary condition -at large distance the escape velocity should tend to zero, i.e., as r → ∞, v 2 → 0. Imposition of this boundary condition is not possible without an explicit expression for the metric functions. Thus we will now concentrate on a class of spacetime, possibly depicting the exterior geometry of a rotating star, in order to determine the escape velocity and hence the Buchdahl bound. Also due to rotation we expect the Buchdahl bound to be different from the equatorial plane versus the corresponding expression at the poles.

Equatorial Buchdahl bound of a rotating star
In this section, we will consider the Buchdahl bound on the equatorial plane of a rotating star. In the absence of any exact exterior solution, we assume the exterior geometry to be of Kerr-like, such that, on the θ = (π/2) plane, the non-zero metric elements, respecting the symmetries of the problem, read, where, Σ 2 = (r 2 + a 2 ) 2 − a 2 ∆ = r 2 (r 2 + a 2 ) + 2M ra 2 . Here M is the mass of the rotating star and a is the specific angular momentum, i.e., a = (J/M ), where J is the angular momentum. Using these metric elements, which are functions of the radial coordinate alone, the expression for velocity of a particle on the equatorial plane, presented in Eq. (20), takes the form, If we now impose the condition that as r → ∞, v 2 → 0, then we must have, E = 1, necessity for an escape velocity. This leads to the following expression for the escape velocity on the equatorial plane from the surface of a rotating star with equatorial radius R as, If this radius R has to represent the Buchdahl radius, our previous analysis suggests that the escape velocity should attain its limiting value (8/9). Thus, in order to find out the Buchdahl limit on the equatorial plane, we impose the condition that v 2 escape ≤ (8/9), and then we obtain the following inequality, Defining, the following two dimensionless parameters -(M/R) ≡ α and (a/M ) ≡ β, the above inequality can be re-expressed in terms of α and β, respectively. However, in general, the above inequality will appear complicated and hence we ignore terms O(β 4 ), O(β 2 α 4 ) and O(β 2 α 3 ), then the inequality simplifies a lot, yielding, Solving the above inequality, quadratic in α, we obtain the following Buchdahl bound on the radius of the rotating Buchdahl sphere on the equatorial plane, Note that for β = 0, i.e., for zero rotation the above bound reduces to (M/R) ≤ (4/9), the expression for spherical stars, as expected. Having derived the corresponding expression on the equatorial plane, we will now derive the Buchdahl limit at the pole, using again the limiting escape velocity.

Axial Buchdahl bound on rotating star
In this section we discuss the radial motion of a particle, barely reaching infinity, escaping along the axis of rotation of the rotating star. This motion corresponds to θ = 0, along with the conserved energy and angular momentum being, E = 1 and L = 0, respectively, characteristics of the radial motion. Thus the escape velocity becomes, where, R is the radius of the rotating star at the axis of rotation. In order to find out the corresponding axial Buchdahl bound, we impose the condition, v 2 escape ≤ (8/9), then we arrive at the following inequality, As above, defining dimensionless quantities, α ≡ (M/R) and β ≡ (a/M ), the above inequality reduces to the following expression in terms of α and β, This implies the following inequality, (8/9)β 2 α 2 + (8/9) − 2α ≥ 0, which is quadratic in both the dimensionless parameters α and β. This can be satisfied, provided, α ≤ α − , where, α − is the lowest route of the algebraic equation (8/9)β 2 α 2 + (8/9) − 2α = 0, in α. Thus we obtain, Note that this bound is very similar to that of a charged shell, expect for the fact that Buchdahl charged shell can exist with (Q/M ) ≤ (3/2 √ 2), however for rotating sphere, the bound is slightly larger, (a/M ) < (9/8). Here also if we have a rotating sphere with 1 < (a/M ) < (9/8), it possible hints towards a naked singular solution upon collapse, provided sufficient amount of rotation is not transferred to other means of energy loss, e.g., gravitational waves.

Equatorial Buchdahl bound for charged rotating shell
As another demonstration, of the ideas presented above, we consider here the Buchdahl bound on the Equatorial radius of a charged and rotating shell, whose outside geometry is governed by the Kerr-Newman spacetime. As in the case of the Kerr spacetime, for the Kerr-Newman spacetime, on the equatorial plane, the relevant metric elements read, where, Here, M is the mass of the black hole and Q is the electric charge. Using these expressions for the metric elements on the equatorial plane in Eq. (20), along with the boundary condition that the motion is radial and the particle barely reaches infinity, the escape velocity takes the form, Here, R is the radius of the rotating charged shell at the equatorial plane. Given the escape velocity, we may impose the condition that v 2 escape ≤ (8/9), in order to obtain the Buchdahl bound, which yields the following inequality, Defining, (M/R) ≡ α, (a/M ) ≡ β and (Q/M ) ≡ γ, the above inequality takes a complicated structure. This can be simplified a lot, if taking a cue from previous results in the context of Kerr, here also we ignore terms O(β 4 ), O(β 2 α 4 ) and similar higher order terms involving γ. Then the inequality becomes quadratic in α, and takes the form, As expected, this demands, α ≤ α − , where α − is the lowest root of the equation obtained by considering equality of the above equation. This yields, We can check that in the limit, β = 0 = γ, we arrive at the Buchdahl bound for a massive star, given by, (M/R) < (4/9). For γ = 0, we get back the bound on the radius for the rotating star, derived in the previous section. Finally, for β = 0, we obtain a value, which is less than Eq. (7). Thus any rotating charged shell satisfying Eq. (35) in the limit β → 0, must also satisfy Eq. (7), therby providing consistency of the computations. We will now present the Buchdahl limit along the axis of a charged rotating shell.

Axial Buchdahl bound for charged rotating shell
As in the case of rotating star, here also we discuss the Buchdahl bound along the axis of the charged rotating shell. The motion is not only axial, i.e., with θ = 0, but also radial, barely reaching infinity, such that E = 1 and L = 0. For this motion the escape velocity along the axis takes the form, where, R is the radius of the charged, rotating shell on the axis. Imposing the condition, v 2 escape ≤ (8/9), we obtain the following inequality, As in the previous section, defining, α ≡ (M/R) and β ≡ (a/M ) and γ ≡ (Q/M ), the previous inequality can be re-expressed in the following manner, This eventually implies the following quadratic equation, (8/9)β 2 α 2 + (8/9) − 2α + γ 2 α 2 ≥ 0, which can be solved to provide the following bound on α, namely, α ≤ α − . Here, α − is the lower root of the above equation, obtained by considering equality, yielding, Note that for β = 0 = γ, we get back the Buchdahl limit for spherical star, while for β = 0, we get back the Buchdahl limit for charged shell and finally for γ = 0, we reproduce the result for the rotating star, derived in the previous section.

Universality of gravitational field energy for Buchdahl sphere
In addition to the universality of the escape velocity and the gravitational potential, as we will demonstrate, there is a universality of the gravitational field energy as well, again associated with the Buchdahl sphere. For this purpose, as in the earlier scenarios involving escape velocity, we consider first a spherically symmetric star of radius r 0 in four dimensional general relativity, whose gravitational field energy is given by [47], which effectively comes from the Brown-York prescription. If the star is now considered to be a Buchdahl sphere with its radius r 0 coinciding with the radius r B of the Buchdahl sphere, such that, (2M/r B ) = (8/9), the gravitational field energy presented in Eq. (40) becomes, On the other hand, the non-gravitational field energy associated with the spherical star in four dimensional general relativity is simply given by, E non−grav = M , which for the Buchdahl sphere reduces to, E Buchdahl non−grav = (4/9)r B . Therefore, it follows that the ratio of the gravitational and non-gravitational energy associated with the Buchdahl sphere becomes, To see that whether this ratio is indeed universal, we now consider the case of a charged shell with radius r 0 , again in the context of four dimensional general relativity. For which, the gravitational field energy becomes [47], Note that for Q = 0, the above expression for gravitational field energy of a charged shell reduces to the gravitational field energy for a massive shell, as depicted in Eq. (40). On the charged Buchdahl shell, r 0 = r B , where the radius r B satisfies the following equation, and hence the gravitational field energy for the charged Buchdahl shell from Eq. (43) becomes, Further, for the spherically symmetric charged shell with radius r 0 , the non-gravitational energy takes the following form [47], and hence it follows that on the Buchdahl sphere the non-gravitational energy becomes, Therefore, the ratio of gravitational and non-gravitational field energy on the charged Buchdahl shell in four dimensional general relativity becomes, This shows that irrespective of the matter content of spherically symmetric Buchdahl sphere, the ratio of the gravitational and the non-gravitational energy is universal for four dimensional general relativity and has a value equal to (1/2). Let us now explore if this universality is respected for generic pure Lovelock theories as well, akin to the case of escape velocity from Buchdahl sphere. The gravitational field energy for a sphere with radius r 0 in pure Lovelock theory can be computed following the Brown-York prescription, which yields [47], On the Buchdahl sphere with radius r B , we have the following relation is being satisfied by the radius, r While, the non-gravitational energy can be determined by the gravitational field energy on the horizon r h , satisfying r For, four dimensional general relativity, we have, d = 4 and N = 1, yielding α = (2/9) and then the above ratio reduces to (1/2), as one can immediately verify. Moreover, even for the charged Buchdahl sphere in pure Lovelock gravity, one can also verify that the above ratio is indeed universal. Thus the ratio of gravitational and non-gravitational energy is universal for Buchdahl sphere irrespective of whether the sphere involves electric charge or not and it holds not only in general relativity but also for pure Lovelock theories of gravity. This further corroborates the fact that the results holding for general relativity also holds in the context of pure Lovelock gravity. Moreover, for a given N , if we choose d = 3N + 1, then, we obtain, α = (2/9) and hence we obtain the maximal value of β = (1/2). This again illustrates the fact that the pure Lovelock gravity of order N in d = 3N + 1 dimensions has properties identical to that of four dimensional general relativity.

Universality of the photon sphere
In this final section, we will demonstrate another universal property of the Buchdahl sphere, namely the ratio of the radius of the Buchdahl sphere to that of the photon sphere in the spherically symmetric context. As a first step towards exploring this universality consider the radius of the photon sphere in static and spherically symmetric geometry of the Schwarzschild spacetime, which becomes, r ph = 3M , such that the ratio of the photon sphere and the Buchdahl sphere becomes, For the case of Reissner-Nordström spacetime, the photon sphere needs to be obtained by solving the algebraic equation rf ′ = 2f , where we have, f (r) = 1 − (2M/r) + (Q 2 /r 2 ). Expanding out the above algebraic equation, it follows that the photon sphere satisfies the equation, which is equivalent to, r 2 − 3M r + 2Q 2 = 0, with the following solution, Note that, in general the above algebraic equation will have two independent solutions, however, the solution within the black hole horizon is not being considered here. For the case of a charged shell, whose outside geometry is Reissner-Nordström, the buchdahl limit is given by [30,31], Therefore, the ratio of the radius of photon sphere and Buchdahl sphere involving the Maxwell charge is given by, Thus we observe that for four dimensional general relativity, the ratio of the radius of the photon sphere and the radius of the Buchdahl sphere is universal, i.e., irrespective of whether the star carries Maxwell charge or, not. Unfortunately, unlike the previous cases, the above universality cannot be generalized for pure Lovelock theories of gravity. This is because, the equation for the photon sphere in N th order pure Lovelock theories of gravity is a higher order algebraic equation, depending on the dimensions of the spacetime and the Lovelock order N , which is difficult to solve for. Thus the ratio of the photon sphere and the Buchdahl sphere cannot be arrived at in a closed form. Thus this universal feature of the Buchdahl sphere holds in four dimensional general relativity alone.

Concluding Remarks
The Buchdahl sphere -a stellar configuration having its radius equal to the Buchdahl limit -has several universal characters, which we have explored in this work. In particular, we have shown that the following properties of a Buchdahl sphere -(a) the escape velocity from the Buchdahl sphere, (b) the gravitational potential on the Buchdahl sphere, (c) the ratio of quasi-local gravitational and non-gravitational energies associated with the Buchdahl sphere and (d) the ratio of the Buchdahl limit with that of the photon sphere -are universal, i.e., independent of the matter content of the star within general relativity. Intriguingly most of these characteristics transcend general relativity and holds for pure Lovelock theories of gravity as well. Using the universality of the escape velocity, we have derived the Buchdahl limit of a slowly rotating stellar configuration -even though no direct computation of the same exists -for the first time in the literature. All of these results boil down to the static and spherically symmetric counterpart in appropriate limits. These universal features of the Buchdahl sphere, along with its compactness, which is of the same order as that of a black hole, implies that the Buchdahl sphere is expected to mimic almost all or most of the features of a black hole spacetime. For example, in terms of the gravitational potential on the horizon (on the Buchdahl limit), we obtain, Φ = 0.5 (Φ = 4/9) in general relativity, which is universal irrespective of the presence of charge and rotation, which arises from the universality of the escape velocity in the respective situations, as v 2 escape = 1 (v 2 escape = 8/9). Moreover, the gravitational redshift at the surface of the Buchdahl sphere always has the value 3, irrespective of charge or, rotation. Since the Buchdahl sphere is as compact as a black hole, but having a non-null boundary, which is therefore accessible to interesting physical scenarios. For example, if a Buchdahl sphere is perturbed, the boundary condition at the surface of the sphere will be different from that of the black hole. In addition, the separation between the photon sphere and the Buchdahl sphere is O(M ), such that the ringdown spectrum of the Buchdahl sphere, under perturbation, will be different from that of a black hole, which can be probed using future gravitational wave detectors. In this sense Buchdahl sphere is an astrophysically more promising and interesting object. It should be also be noted that all of these universal features follow entirely from the exterior solution without any reference to interior matter distribution and geometry. This is exactly as is the case for black hole and hence would be accessible to astrophysical observations. For instance, if an object is observed, with its gravitational redshift at the surface being larger than 3, it would strongly indicate that it is more compact that a Buchdahl sphere, and without any exotic matter support, is possibly a black hole. Thus the surface red shift could serve as a good practical way of distinguishing the Buchdahl sphere from a black hole or, for that matter from exotic compact objects.
Another interesting aspect associated with the Buchdahl sphere is the result that the extremal limit for a Buchdahl sphere is over extremal relative to a black hole spacetime. For instance, in the case of a charged Buchdahl sphere, the extremal limit corresponds to (Q 2 /M 2 ) = (9/8) > 1. Thus it is worthwhile to ask, whether it will be possible to form an extremal charged BH by starting from a near extremal Buchdahl sphere with 1 < (Q 2 /M 2 ) < (9/8) and then throwing in the neutral matter. This could perhaps be the only way to form an extremal black hole, if at all. Further, it is also known that non-extremal black hole cannot be extremalized by adiabatic test particle accretion, following which we wish to study whether a similar situation holds for Buchdahl sphere as well in a future work. In this connection, it is also of interest to probe whether a Buchdahl sphere can be over-extremalized and whether the results similar to the weak cosmic censorship conjecture for black holes hold good for Buchdahl sphere as well.