McVittie solution in f(T) gravity

We show that McVittie geometry, which describes a black hole embedded in a FLRW universe, not only solves Einstein equations but also remains as a non-deformable solution of f(T) gravity. This search for GR solutions that survive in f(T) gravity is facilitated by a null tetrad approach. We also show that flat FLRW geometry is a consistent solution of f(T) dynamical equations not only for $T=-6H^{2}$ but also for $T=0$, which could be a manifestation of the additional degrees of freedom involved in f(T) theories.


II. TELEPARALLEL FRAMEWORK
Teleparallelism is a framework to describe gravity where the role of the metric tensor g µν is taken by the tetrad or vierbein {e a (x µ )}, a = 0, 1, 2, 3. The tetrad is a set of four vectors at each point of the spacetime which are linked to the metric by the condition of orthonormality e a · e b = η ab , where η ab =diag(1, −1, −1, −1) is the Minkowski symbol. In coordinate bases, the tetrad and its dual co-tetrad {e a (x µ )} write as e a = e µ a ∂ µ and e a = e a µ dx µ , where e µ a and e a µ are the components of the tetrad and its inverse which accomplish duality relationships (e a µ e µ b = δ a b and e a µ e ν a = δ ν µ ). By combining duality and orthonormality, one can get the metric tensor of the manifold in terms of a tetrad in the tangent space, Of course, this relationship is invariant under (local) Lorentz transformations of the tetrad. As seen, we have used Greek indices to label coordinates and tensor components in coordinate basis; instead Latin indices a, b, ... = 0, 1, 2, 3 label the vectors taking part in the tetrad, and tensor components with respect to these Lorentzian frames. These indices are lowering and raising by metric tensor and the Minkowski symbol respectively. It is well known that General Relativity can be reformulated by using the tetrad as the dynamical variable. In fact, in the so-called Teleparallel Equivalent of General Relativity (TEGR) [6][7][8] the action is built from the object of anholonomity de a , which reads de a = e a µ T µ ρν dx ρ ∧ dx ν in a coordinate basis, where de a coincides with the torsion T a = de a + ω a b ∧ e b when the spin connection ω a b is chosen to be zero. Such choice is the Weitzenböck connection [51], which also implies that the curvature R a b = dω a b + ω a c ∧ ω c b vanishes. According to Eq. (2) the Christoffel symbols for the Weitzenböck connection are Weitzenböck connection leads to the vanishing of the covariant derivative of the tetrad, which also means that this connection is metric-compatible (see Eq. (1)). This property defines an absolute parallelism in the spacetime. In fact, W ∇e µ a ≡ 0 means that a parallel transported vector keeps constant its projections on the tetrad, irrespective of the path as a consequence of the zero curvature. This is the reason why the name teleparallelism comes up. As it can be seen, Weitzenböck torsion (2) is invariant only under global Lorentz transformations of the tetrad. So, Weitzenböck geometry selects a preferred global frame modulo global Lorentz transformations, in spite that the metric is invariant under local Lorentz transformations.

A. The Teleparallel Equivalent of General Relativity
The TEGR action is where κ = 8πG, e = det[e a µ ] = √ −g and the torsion scalar or Weitzenböck invariant T is defined by the contraction of the torsion tensor ((2) and the superpotential, that is tagged as the contorsion tensor. Then, the torsion scalar can be expressed by the following quadratic combination of the components of the torsion tensor One can compute the Levi-Civita scalar curvature R in terms of the tetrad by using Eq. (1); then it is obtained a relation between T and R which shows that both the Einstein-Hilbert Lagrangian only differs from the TEGR Lagrangian in a four-divergence, Hence the equations of motion are fully equivalent, showing the equivalence between GR and TEGR pictures. Although the tetrad field has 16 independent components, in contrast with the 10 independent components of the symmetric metric tensor, both TEGR and GR have the same number of degrees of freedom, as it is expected from their equivalence. In fact, Eq. (6) implies that TEGR only governs the dynamics of the metric, which is invariant under local Lorentz transformations of the tetrad. Then, the tetrad is only determined modulo this local symmetry group. Therefore, the extra components of the tetrad does not represent new dynamical degrees of freedom. In fact, the local Lorentz invariance of the relation (1) means that the metric is related with an infinite set of tetrad fields, connected by local Lorentz transformations. Note also that teleparallel vacuum solutions do not compel T to vanish, as opposite to R in GR. This point is clearly shown in Eq. (6) where R = 0 implies that T is a four-divergence.

B. f (T ) gravity
In analogy with f (R) gravity [52][53][54][55][56][57], where the GR Lagrangian is extended to an arbitrary function f of the curvature scalar R, f (T ) gravity is obtained by replacing the TEGR Lagrangian with an arbitrary function f of the torsion scalar T [12][13][14] The dynamical equations of f (T ) gravity are computed by varying the modified teleparallel action with respect to the tetrad, yielding where T ν λ is the energy-momentum tensor. Of course, TEGR dynamics is recovered if f (T ) = T . Remarkably, it is straightforward to verify that the equations of motion are of second order with respect to the tetrad (just as in GR with respect to the metric), since T ρ µλ and S µν ρ are linear in first derivatives of the tetrad. Instead, other alternative theories of gravity have dynamical equations of higher order.
From the equivalence relation expressed in Eq. (6), it is manifest that f (T ) gravity is a non-local Lorentz invariant theory; by extending the TEGR Lagrangian to a function f (T ), the four-divergence (non-invariant) term remains encapsulated inside the function f . 1 While TEGR is a theory for the metric (as GR is), f (T ) gravity is a theory for the tetrad. This is so because the modified teleparallel dynamical equations are not invariant under local Lorentz transformations but only under global Lorentz transformations. 2 Thus the dynamical equations possess information exclusively associated to the tetrad field. This means that f (T ) theories dynamically endow the spacetime with an absolute parallelism. This also means that if looking for a solution associated with a given metric, then the symmetries of the metric are not enough to anticipate the form of the tetrad solving the equations. For instance, it was shown that Schwarzschild geometry [26] and non-flat FLRW spacetimes [27] require non-trivial tetrads in order to consistently solve the equations of motion. Therefore the gravitational field is encoded in a set of preferred reference frames, which should not depend on the function f considered [32]. In this line, it was proposed that a set of suitable frames would be defined up to certain local Lorentz transformations, which are a remnant symmetry group which depends on the specific spacetime considered [37]. In fact, in a mathematical context, it has been known for a long time that the vector fields capable to parallelize a manifold are not unique (see for instance, Ref. [68]).
In the present work, we rely on results displayed in [26,36] where Schwarzschild and Kerr spacetimes proved to remain as vacuum solutions in f (T ) gravity, since both geometries admit a tetrad where the torsion scalar vanishes (T = 0) or is constant (T = T c ). In fact if ∂ µ T = 0, Eq. (8) can be arranged as follows where we define G ν η from Eq. (8) by taking f (T ) = T , 1 The equivalence between GR and TEGR is broken after the generalization procedure; therefore f (T ) gravity is not at all equivalent to f (R) gravity. 2 It should be noted too that the local Lorentz symmetry could be broken at the level of quantum gravity (Planck scale) and many candidate theories of quantum gravity predict at some point the loss of local Lorentz symmetry. In fact, there are many attempts to examine in detail the range in which the Lorentz symmetry is preserved [58][59][60][61][62][63]. Other alternative descriptions of gravity in vogue nowadays such that Hoȓava-Lifshitz [64,65] and AEther-Einstein [66,67] are not local Lorentz invariant in the time sector.
(G (µν) is the Einstein tensor). Remarkably the dynamical equations (9), when the energy-momentum tensor is symmetric, are nothing but (TEGR) Einstein equations with a scaled Newton constantG = G/f ′ (T c ) and cosmological Therefore, if f (T = 0) = 0 then the TEGR solutions having T = 0 remain as solutions of f (T ) gravity (in the case of non-vacuum solutions we should adjust the Newton constant). Even if T c = 0 one could still regard any TEGR solution having T = T c as a solution to f (T ) equations for proper values of cosmological and Newton constants. In the following sections we will take advantage of this property to figure out whether a metric solving Einstein's equations could survive as a solution of f (T ) equations or not. Following the strategy displayed in [36], we will exploit the local Lorentz invariance of TEGR to look for solutions having T = 0 by starting from a known solution. If we success, then we will state that the so obtained solution remains as a solution to f (T ) gravity. The null tetrad approach will be very useful for this purpose.

III. MCVITTIE GEOMETRY
McVittie geometry [47] describes a cosmological black hole, which means a black hole solution embedded in an expanding FLRW universe. A comprehensive review of McVittie geometry was carried out by Nolan in the late nineties [48]. Nonetheless, Kaloper et al. [50] went one step further by analyzing some misconceptions in the geometrical interpretation of the solution. The original McVittie spacetime assumes vanishing (Levi-Civita) spatial curvature in the asymptotically FLRW region, but also it can be easily generalized to include positive or negative spatial curvature [49]. Here we will focus on the spatially flat case. It is assumed that the spatial curvature of the FLRW does not significantly influence the dynamics around the central mass, accounting for that the radius of curvature is (much) greater than the gravitational radius of the mass source. Then, the standard McVittie geometry is described by the metric where µ = m(2a(t)|x|) −1 , with m/G the mass of the source, a(t) is the asymptotic cosmological scale factor, and the center of the spherical symmetry is at x = 0. This is an exact solution to Einstein's equations for an arbitrary mass m provided that a(t) solves the Friedmann equation. As might be expected, if m = 0 the geometry corresponds to the standard flat FLRW spacetime, and if a(t) = 1 the line element describes the Schwarzschild solution (in isotropic coordinates). For a ∼ e H0t , the metric reduces to the case of Schwarzschild-de Sitter (by adopting a positive cosmological constant). It can be verified that McVittie solution has a spacelike and inhomogeneous singularity at µ = 1 (which means that a(t)|x| = m/2) where the surface lies in the causal past of all spacetime events so that it should be properly interpreted as a cosmological big-bang singularity (see Ref. [50] for more detail).
In analogy to the FLRW case for a perfect fluid, the energy density scales with the scale factor a(t) and commands the expansion rate where H(t) =ȧ(t)/a(t) is the Hubble parameter. Remarkably, the energy density is constant along slices where t is constant, but the pressure on fixed t slices is not homogeneous The inhomogeneous pressure constitutes the necessary non-gravitational balancing force (when the mass is constant and the energy density is spatially homogeneous) to compensate the gravitational attraction of the central mass.
The coordinates employed in the line element given by Eq. (11) are such that |x| covers the exterior of the black hole twice, that is m/2 < |x| < ∞ covers the same exterior region as 0 < |x| < m/2. Therefore, Kaloper et al. [50] propose another coordinate choice which in turn imitates better the familiar static form of the Schwarzschild metric. The new radial coordinate is defined by where |R| = R represents the "spherical area" coordinate. Since the relation between a(t)|x| and R is quadratic, the coordinate transformation (14) actually defines two separate branches The physically relevant branch is the one with the (−) sign, since R → ∞ implies |x| → ∞, therefore the geometry is asymptotically FLRW-like. By applying the transformation defined in (14), the McVittie metric becomes where dΩ 2 = dθ 2 + sin 2 θ dφ 2 . It is now evident that a constant value of H leads to the Schwarzschild-de Sitter metric in coordinates which are analogous to outgoing Eddington-Finkelstein coordinates.

IV. MCVITTIE SOLUTION IN f (T ) GRAVITY
As previously mentioned, f (T ) gravity is not locally Lorentz invariant. This means that tetrads connected by local Lorentz transformations, which reproduce the same metric tensor, are not equivalent at the level of the equations of motion of f (T ): not all of them will represent a proper parallelizing field of frames (in the sense of being a consistent solution of the dynamical equations). We can state that f (T ) gravity selects the parallelization of the spacetime. Finding suitable tetrad solutions is then quite awkward in f (T ) theory, since the symmetry of the searched metric is not enough to determine the form of the tetrad. However, as explained at the end of Section II, one can exploit the local Lorentz invariance of TEGR solutions to force the scalar torsion T to be zero. If such a purpose is attainable, we will obtain a tetrad solving f (T ) gravity equations too (whenever f (T = 0) = 0). This also means that the same metric solving TEGR equations is admisible in f (T ) gravity. Since the condition T = 0 is not affected by global linear transformations of the tetrad, it could be easier to look for the condition T = 0 by using a null tetrad [36]: given an orthonormal tetrad {e a } = {e 0 , e 1 , e 2 , e 3 }, a null tetrad can be defined as {n a } is a null basis, n a · n a = 0, but it is not orthogonal (l · n = −m · m = 1). Then, Eq. (1) can be rewritten in this new basis as where η ab now reads In terms of the tensorial product of the elements of the null tetrad the metric reads g = n ⊗ l + l ⊗ n − m ⊗ m− m ⊗ m. A local Lorentz boost along the direction of e 1 , with parameter γ( n}, which clearly does not modify the form of the metric tensor. The null tetrad approach to get T = 0 exploits the freedom in the choice of the function λ(x µ ). The strategy is the following: (i) determine the null tetrad {n a }, (ii) apply the transformation in the {l, n}-sector, (iii) impose the condition that the torsion scalar T be zero. This approach was successfully implemented for the Kerr geometry [36]; here we will show that it is also useful in McVittie spacetime. 3 Then, we compute the null tetrad associated to the metric given in Eq. (16) with coordinates (t, R, θ, φ) and we perform a radial boost yielding where the function λ has to be determined. For this null tetrad, we determine the torsion scalar by means of Eq. (4): By imposing that the torsion scalar vanishes, we solve the differential equation to obtain Note that λ allows for an additive function of (R, θ) without affecting the result T = 0. In summary, according to Eq. (9), we conclude that f (T ) gravity is not able to deform McVittie metric since we have just found a solution leading to a vanishing torsion scalar which consistently solves the dynamical equations (notice the scaling of Newton constant κ), where, due to the non-diagonal form of the metric tensor (16), the stress-energy tensor for a perfect fluid becomes with ρ and p the energy density and the pressure, respectively [49].

V. COSMOLOGY WITH T = 0
As it has been extensively studied in the literature, flat FLRW cosmology accepts the naive diagonal tetrad (in the Cartesian chart) as a suitable solution in the context of f (T ) gravity, which leads to T = −6H 2 (t) [12,27]. However, McVittie metric for m = 0 reduces to FLRW metric. Therefore, we have also obtained the outstanding result that there exists a a tetrad {e a } having T = 0 in flat FLRW spacetime, from which immediately follows that that such tetrad is also a solution f (T ) gravity (assuming f (T = 0) = 0, and taking care of the scaling of Newton constant). By replacing m = 0 in Eq. (20), we get that the null tetrad associated with {e a } is where λ(t, R) is the function given in Eq. (22). According to Eq. (24), the dynamical equations for m = 0 become This remarkable result will be deeply studied in a forthcoming article [69]. At first sight, solutions providing the same metric tensor but different torsion scalars suggest the involvement of the extra degrees of freedom characteristic of f (T ) gravity. They constitute differrent admisible parallelizations of FLRW geometry.
To compare both f (T ) solutions, firstly, we will get the orthonormal tetrad e a µ associated with the null tetrad given in Eq. (26), and then we will write it in the original chart (t, |x|). In fact, by performing the inverse coordinate transformation of Eq. (14), which reduces to R = a(t) |x| = a(t) r for m = 0, we will get e a µ ′ , in terms of the spherical radial coordinate r. The first step is achieved through a transformation L a b such that (see Eq. (17)) where we set λ ≡ λ(t, R) to abbreviate the notation. In the second step, we will perform the coordinate transformation x µ = (t, R, θ, φ) −→ x µ ′ = (t, r, θ, φ). The tetrad {e a } is a geometrical object independent of the coordinate choice, but its components e a µ will change. According to Eq. (14), the 1-forms dx µ and dx µ ′ will not change except for dR =ȧ r dt + a dr. Then, it is obtained that Tetrad (31) is the one having T = 0, as written in usual spherical coordinates. On the other hand, there exist a tetrad which has T = −6H 2 ; this tetrad is diagonal in the Cartesian chart, but in spherical coordinates reads e a ′ µ ′ =    1 0 0 0 0 a(t) sin θ cos ϕ a(t) r cos θ cos ϕ −a (t) r sin θ sin ϕ 0 a(t) sin θ sin ϕ a(t) r cos θ sin ϕ a(t) r sin θ cos ϕ 0 a(t) cos θ −a(t) r sin θ 0 Of course it should exist a local Lorentz transformation connectin the tetrads (31) and (32), since both describre the same FLRW metric. In fact, the local Lorentz transformation is i.e. e a ′ = Λ a ′ b e b , which separates as the product of two rotations and one boost In Ref. [37] it was shown that each solution to f (T ) dynamical equations can allow for a set of local Lorentz remnant symmetries. Such symmetries are generated by Lorentz transformations accomplishing the condition Our pair (e a , Λ a ′ b ) does not satisfy this relationship. This is because, although both tetrads have consistent equations of motion, their associated torsion scalars are different; so the transformation (33) is not a symmetry of the action. Also, they have different classification concerning to the n-closed-area-frame (n-CAF), distinction firstly introduced in the same paper. Remember that a solution of f (T ) gravity is n-CAF if n of the six pairs (e a , e b ) satisfy the equation d(e a ∧ e b ) = 0. (36) In this particular case, it can be proven that the tetrad (31) is a 1-CAF, since the only combination that is zero is the one with (e 0 , e 1 ). On the other hand, the tetrad (32) is a 3-CAF [37]. This means that the second one allows for three independent local Lorentz transformations leaving T unchanged, while the first solution admits only the local boost associated with the remnant freedom of the function λ. On the contrary, the transformation (33) is not a remnant symmetry of the f (T ) dynamical equations because it changes the torsion scalar T .

VI. CONCLUSIONS
The main purpose of f (T ) gravity would be finding low/high-energy deformations of Einstein gravity to solve its shortcomings in a geometric framework. However, the null tetrad approach shows that it is rather easy to get TEGR solutions with constant torsion scalar. These solutions survive in f (T ) gravity, associated with modified Newton and cosmological constants. This is the way we followed to show that McVittie geometry is a solution to f (T ) gravity. Remarkably, by taking m = 0 we also obtained a new consistent solution for FLRW universe in f (T ) gravity, which has T = 0. To the best of our knowledge, this is the first time that a consistent cosmological solution with a vanishing torsion scalar is introduced in the literature. The fact that the torsion scalar differs from −6H 2 could be a manifestation of the extra degrees of freedom of the theory, which is right now a topic under consideration [69].