Photon Sphere and Perihelion Shift in Weak F(T) Gravity

PHYSICAL REVIEW D 100, 084064 (2019)

Photon sphere and perihelion shift in weak fðTÞ gravity

† ‡ Sebastian Bahamonde ,1,2,* Kai Flathmann,3, and Christian Pfeifer1, 1Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia 2Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom 3Department of Physics, University of Oldenburg, 26111 Oldenburg, Germany

(Received 9 August 2019; published 29 October 2019)

Among modified theories of gravity, the teleparallel fðTÞ gravity is an intensively discussed model in the literature. The best way to investigate its viability is to derive observable predictions which yield evidence or constraints for the model, when compared with actual observations. In this paper we derive the photon sphere and the perihelion shift for weak fðTÞ perturbations of general relativity. We consistently calculate first order teleparallel perturbations of Schwarzschild and Minkowski spacetime geometry, with which we improve and extend existing results in the literature.

DOI: 10.1103/PhysRevD.100.084064

I. INTRODUCTION arbitrary f is a difficult task. The main challenge is to find the tetrad, to which one can consistently associate a With the observation of gravitational waves of merging vanishing spin connection. This tetrad does not only have black holes [1] and the first direct picture of a black hole to satisfy the symmetric part and antisymmetric part shadow in the center of the galaxy M87 [2], the possibilities (the spin-connection part) of the field equations [10] but to observe the behavior of gravity in the strong field regime has increased enormously. The newly obtained data is the also must yield a torsion scalar which vanishes for the perfect basis to understand the viability range of general Minkowski spacetime limit. A further subtlety is that some solutions of the fðTÞ field equations yield a constant relativity and possible modified gravity theories, suggested ð Þ as its generalization. Here we derive the influence of a torsion scalar T. In this case f T gravity is identical to teleparallel modification of general relativity on the photon TEGR plus a cosmological constant and nothing new is sphere of black holes and on the perihelion shift of elliptic obtained. The latter feature is for example present in the orbits in spherical symmetry. This is a step to investigate first study which tried to find spherically symmetric the influence of teleparallel gravity on more realistic solutions [11] and also in a later study which used the ’ spinning black holes with axial symmetry. Noether s symmetry approach to find solutions [12]. Teleparallel theories of gravity are formulated in terms of There are only a few publications deriving exact sol- a tetrad of a spacetime metric and a spin connection, instead utions with the correct field equations (see e.g., [13]). In of in terms of a spacetime metric and its Levi-Civita addition, some regular black hole solutions (perturbatively connection [3]. This structure allows for the construction and exact) have been found correctly in [14,15]. When one of a huge variety of theories of gravity beyond general considers matter, there are some works which have studied relativity, among them the most famous model, the so- the possibility of constructing stars or wormhole solutions called fðTÞ gravity [4,5]. In this theory, the Lagrangian is in different teleparallel theories of gravity [16–24]. Overall given by an arbitrary function f of the torsion scalar T, the issue of finding exact spherically symmetric solutions in which defines the teleparallel equivalent formulation of fðTÞ gravity is still an open problem. general relativity (TEGR). Numerous viability criteria for Instead of looking for full analytical solutions, an fðTÞ gravity have been derived in the context of cosmology alternative way to study the astrophysical effects of [6–9]. However, not much work has been done in spherical modified theories of gravity is to employ perturbation and axial symmetry, mostly due to the lack of analytic theory. The influence of deviations from TEGR can be solutions of the field equations. To solve the fðTÞ gravity investigated, by setting fðTÞ¼T þð1=2ÞϵαTp. This field equations in spherical symmetry in all generality for model contains TEGR (and GR) in the limit the perturbation parameter ϵ or the coupling constant α goes to zero. It ϵ ≪ 1 *[email protected] is assumed that the deviation from TEGR is small, ( ), † [email protected] and hence, only first order terms in ϵ are relevant in all ‡ [email protected] calculations. In this paper, we consider perturbations

a μ around two different background geometries: Minkowski Throughout the paper we denote h μ and ha for the spacetime and Schwarzschild spacetime. tetrad and its inverse, respectively, where Latin indices refer For the first case we keep the exponent p>1=2 and to tangent space indices and Greek to spacetime indices. solve the spherically symmetric perturbative fðTÞ field Our signature convention is ðþ; −; −; −Þ and we work in equations in vacuum. We find that, to first order, the units where G ¼ c ¼ 1. teleparallel perturbation of general relativity has no influence at all. This finding is in conflict with results found II. COVARIANT FORMULATION OF fðTÞ earlier in [25]. A problem with these earlier derivations is GRAVITY IN SPHERICAL SYMMETRY that the fðTÞ field equations presented in [25], Eqs. (8) through (10), do not have Schwarzschild geometry as a Throughout this paper we employ the covariant formu- solution for fðTÞ¼T and vanishing matter. Moreover the lation of teleparallel gravity [27] in the Weitzenböck gauge, tetrad used there does not yield a torsion scalar which also called the pure tetrad formalism. That means we vanishes in the Minkowski spacetime limit. These short- consider a tetrad, its torsion, and a vanishing spin con- comings made us redo the calculations, while paying nection. All degrees of freedom are encoded in the tetrad particular attention to the consistency of the perturbation which, in the end, solves the symmetric and the antisym- ð Þ theory. metric part of the f T field equations, and yields a For the second case, the first order field equations are vanishing torsion scalar in the Minkowski spacetime limit. more involved and cannot be solved for general p. To find We would like to stress that this is as equivalent as the astrophysical impact of the parameter p, we derive considering a nonvanishing spin connection and another perturbative solutions for p ¼ 2 to 10, from which we tetrad, which together solve the symmetric and antisym- – calculate the circular particle orbits for massless particles metric parts of the field equations [27 29]. and the perihelion shift for nearly circular massive particle orbits. The circular orbits of massless particles define the A. Covariant teleparallel gravity photon sphere, which is interesting in particular, since it The fundamental variables in teleparallel theories defines the edge of the shadow of the black hole. For p ¼ 2 of gravity are the tetrad of a Lorentzian metric a b the perihelion shift has already been studied in the literature g ¼ ηabθ ⊗ θ , which can be expressed in local coordi- and we recover the results from [26]. Comparing our nates as calculation with the previous one demonstrates explicitly a a μ μ that the covariant formulation of teleparallel gravity works θ ¼ h μdx ;ea ¼ ha ∂μ; well. We employ a vanishing spin connection and a a a a b θ ðe Þ¼δ ⇒ gμν ¼ η h μh ν; ð1Þ nondiagonal tetrad, while in [26] a nonvanishing spin b b ab connection a diagonal tetrad was used. and a flat, metric compatible spin connection that is The main aim of this paper is to present a careful generated by local Lorentz matrices Λa derivation of the first order fðTÞ field equations in the b single tetrad framework for the models mentioned above a a a −1 c a b ω μ ¼ ω μðΛÞ¼Λ ∂μðΛ Þ ; η Λ Λ ¼ η ; and to derive the perturbative solutions around Minkowski b b c b ab c d cd and Schwarzschild geometry. Eventually this procedure ð2Þ gives insights about the phenomenological consequences of the teleparallel corrections to general relativity. This work which poses torsion prepares a more general study where we will derive the a ¼ 2ð∂ a þ ωa b Þ ð Þ phenomenological consequences of teleparallel perturba- T μν ½μh ν b½μh ν : 3 tions of Kerr geometry, with and without cosmological constant. In the Weitzenböck gauge, the spin connection is set to be ωa ¼ 0 The article is structured as follows: In Sec. II we give an zero ( bμ ) and so, the torsion tensor reduces to a a overview about the covariant formulation of fðTÞ gravity T μν ¼ ∂½μh ν. From here on we will work in the and then, we find the corresponding field equations for any Weitzenböck gauge. This is equivalent to having a non- spherically symmetric spacetime. Section III is devoted to vanishing spin connection and a tetrad which solve their studying the weak power-law fðTÞ model for perturbations respective antisymmetric part of the field equations. around Minkowski and Schwarzschild geometries to find A detailed discussion about this equivalence can be found the correct metric coefficients which solve the first order in Refs. [27–29]. field equations. The particle motion phenomenology for the The teleparallel equivalent of general relativity is con- squared power-law fðTÞ case for the Schwarzschild back- structed from the action ground is studied in Sec. IV, deriving the deviation from Z 1 TEGR (or GR) of the photon sphere and the perihelion ¼ 4 j j þ L ð ΨÞ ð Þ STEGR d x h 2 T m g; ; 4 shift. We conclude our main results in Sec. V. 2κ

084064-2 PHOTON SPHERE AND PERIHELION SHIFT IN WEAK FðTÞ … PHYS. REV. D 100, 084064 (2019) pffiffiffiffiffiffi 2 a where κ ¼ 8π, jhj¼detðh μÞ¼ −g is the determinant rewritten purely in terms of spacetime indices by contrac- a L ð ΨÞ gμρ h σ of the tetrad, m g; is the matter Lagrangian for matter tion with and to take the form minimally coupled to gravity via the metric generated by the tetrads, and the so-called torsion scalar T reads as 1 2 Hσρ ¼ κ Θσρ: ð Þ follows: 2 8

a μν T ¼ T μνS Their symmetric part is sourced by the energy-momentum a 1 1 tensor, while their antisymmetric part is a vacuum con- ¼ σ ρμ ν þ 2 ρ σμ ν þ η μρ νσ a b 2 ha g hb hb g ha 2 abg g T μνT ρσ: straint for the matter models we consider. The latter is equal to the variation of the action with respect to the flat spin- ð5Þ connection components [28,29], μν μν ¼ 1 ð μν − The superpotential Sa is given by Sa 2 K a 1 2 μ λν ν λμ HðσρÞ ¼ κ ΘðσρÞ;H½σρ ¼ 0: ð9Þ ha Tλ þ ha Tλ Þ in terms of the contortion tensor 2 μν 1 νμ μν μν K a ¼ 2 ðT a þ Ta − T aÞ, and the appearing components of the metric are understood as a function of the The explicit form of these equations can be found for tetrads. example in Eqs. (26) and (30) in [30] by setting the scalar The modified teleparallel theory of gravity we are field ϕ to zero. We do not display these here since we will investigating is fðTÞ gravity, which is a straightforward derive the spherically symmetric field equations directly generalization of the action (4) as follows: from (7). Z 4 1 S ð Þ ¼ xjhj fðTÞþL ðg; ΨÞ : ð Þ B. Spherical symmetry in fðTÞ gravity f T d 2κ2 m 6 In this section, the fðTÞ field equations for a spherically The function f is an arbitrary function of the torsion scalar. symmetric spacetime will be derived. Let us first start with a Variation with respect to the tetrad h μ yields the field the following spherically symmetric metric in standard equations [27] spherical coordinates ðt; r; θ; ϕÞ: 1 1 2 ¼ 2 − 2 − 2ð θ2 þ 2θ ϕ2Þ ð Þ μ b μν μν μν ds Adt Bdr r d sin d ; 10 fðTÞh þ f T ν S þ ∂νðhS Þ þ f S ∂νT 4 a T a b h a TT a 1 where A ¼ AðrÞ and B ¼ BðrÞ are positive functions which ¼ κ2Θ μ ð Þ 2 a ; 7 depend on the radial coordinate. This means we consider the outside region of possible black holes. To calculate the μ with Θa being the energy-momentum tensor of the matter field equations, we employ the following off-diagonal 2 2 field, fT ¼ ∂f=∂T and fTT ¼ ∂ f=∂T . They can be tetrad [16]:

0 pffiffiffiffi 1 A 000 B pffiffiffiffi C B C 0 B cosðϕÞ sinðθÞ r cosðϕÞ cosðθÞ −r sinðϕÞ sinðθÞ a B C h ν ¼ B pffiffiffiffi C: ð11Þ B 0 B sinðϕÞ sinðθÞ r sinðϕÞ cosðθÞ r cosðϕÞ sinðθÞ C @ pffiffiffiffi A 0 B cosðθÞ −r sinðθÞ 0

This tetrad together with a vanishing spin connection consistently defines a spherically symmetric teleparallel geometry. Hence, it is consistent to derive the fðTÞ field equations (7) from this tetrad with vanishing spin connection. Equivalently one could choose a diagonal tetrad with a nonvanishing spin connection [31], as was done in [26]. For this setup the torsion scalar becomes pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2ð BðrÞ − 1ÞðrA0ðrÞ − AðrÞ BðrÞ þ AðrÞÞ T ¼ − : ð12Þ r2AðrÞBðrÞ

Clearly, if A → 1 and B → 1 (Minkowski limit), the torsion scalar vanishes. Calculating the field equations (7) contracted a 0 with h σ and an anisotropic fluid energy-momentum tensor, defined by the energy density ρ ¼ Θ 0, the radial and the lateral

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1 2 pressures −pr ¼ Θ 1 and −pl ¼ Θ 2 respectively, we find the nonvanishing independent spherically symmetric fðTÞ field μ equations are the diagonal components H μ (no sum taken), pffiffiffiffi pffiffiffiffi 1 rBð B − 1ÞA0 þ AðrB0 þ 2B3=2 − 2BÞ ð B − 1Þ 1 κ2ρ ¼ f þ T0f þ f; ð13Þ 2 2r2AB2 T rB TT 4 pffiffiffiffi pffiffiffiffi 1 rð B − 2ÞA0 þ 2Að B − 1Þ 1 κ2p ¼ − f − f; ð14Þ 2 r 2r2AB T 4

1 −r2BA02 þ rAð−rA0B0 − 4B3=2A0 þ 2BðrA00 þ 3A0ÞÞ þ A2ð−2rB0 − 8B3=2 þ 4B2 þ 4BÞ κ2p ¼ f 2 l 8 2 2 2 T pffiffiffiffi r A B rA0 − 2Að B − 1Þ 1 þ T0f − f; ð15Þ 4rAðrÞBðrÞ TT 4 where primes denotes derivatives with respect to the radial In each instance we will compare our results and the ones coordinate. There are only three independent equations obtained in [25,26]. ϕ θ since H ϕ ∼ H θ. This also demonstrates that the tetrad (11) is a so-called good tetrad, i.e., it solves the antisym- III. WEAK POWER LAW fðTÞ GRAVITY metric field equations H½μν ¼ 0 with vanishing spin con- In this section we turn our focus to a power-law fðTÞ nection. We like to remark here that our choice of tetrad is model that reads as not the only good tetrad in this sense. The tetrad presented in [10] could be chosen equally well from this point of 1 ð Þ¼ þ ϵα p ð Þ view. However not all of these good tetrads, in the sense of f T T 2 T ; 16 the field equations, yield consistently a torsion scalar that vanishes for the Minkowski spacetime limit. where α and p are constants and ϵ ≪ 1 is a small tracking A consistency check of the field equations is to set parameter that, similarly as was done in [26], is used to ð Þ¼ ¼ 1 ¼ 0 f T T, hence fT and fTT , and to see if for make the series expansion in a coherent way. It allows us ρ ¼ ¼ ¼ 0 ð Þ¼ pr pl , Schwarzschild geometry [A r to easily track quantities that are considered to be small −1 BðrÞ ¼ 1–2M=r] solves the field equations, which is throughout our calculations. the case. Since we are interested in perturbations of a The last remark leads us to the point that the equations we Schwarzschild background, the ansatz we employ for the derived above differ from the spherically symmetric fðTÞ metric coefficients is field equations in [25]. First, their field equations (8)–(10) are not solved by Schwarzschild geometry for fðTÞ¼T 2M AðrÞ¼1 − þ ϵaðrÞ; ð17Þ and ρ ¼ pr ¼ pl ¼ 0, as it is the case for our equations. r Second the off-diagonal tetrad they choose has the problem 2M −1 that its torsion scalar does not vanish for the Minkowski BðrÞ¼ 1 − þ ϵbðrÞ; ð18Þ spacetime limit, which means that it is not the correct tetrad r to which one associates a vanishing spin connection. We will now solve the fðTÞ field equations for a specific where aðrÞ and bðrÞ are functions of the radial coordinate. ð Þ power-law choice for f to first order in a perturbation If one uses the above metric coefficients in the f T power- – around Minkowski and Schwarzschild spacetime geometry. law spherically symmetric field equations (13) (15) and then expands up to first order in ϵ, the equations become

1 ð−1Þpþ12p−3ðp − 1Þ μ2ðμ2 − 2ÞbðrÞ μ4b0ðrÞ κ2ϵρ ¼ ϵ α ðμ − 1Þ2p−1ðμ − 1 þ pð1 þ μð2 þ 5μÞÞÞ − þ ; ð19Þ 2 ðr2μÞp 2r2 2r 1 ð−1Þpþ12p−3ðp − 1Þ a0ðrÞ ðμ2 − 1ÞaðrÞ μ2bðrÞ κ2ϵp ¼ ϵ −α þ þ − ; ð20Þ 2 r ðr2μÞp 2r 2μ2r2 2r2

084064-4 PHOTON SPHERE AND PERIHELION SHIFT IN WEAK FðTÞ … PHYS. REV. D 100, 084064 (2019) 1 ð−1Þpþ12p−5ðp − 1Þ κ2ϵp ¼ ϵ −α ðμ − 1Þ2pðp þ 2ð2 þ pÞμ þ 5pμ2Þ 2 l μðrμÞp 1 ð3μ2 − 1Þa0ðrÞ ðμ4 − 1ÞaðrÞ μ2ðμ2 þ 1Þb0ðrÞ ðμ4 − 1ÞbðrÞ þ a00ðrÞþ − − þ ; ð21Þ 4 8μ2r 8μ4r2 8r 8r2 where μ ¼ð1 − 2M=rÞ1=2 was introduced for simplicity and we assumed that the energy-momentum tensor is zero to zeroth order in ϵ. The latter assumption is necessary in order to have Schwarzschild geometry as consistent zeroth order solution. As usual in perturbation theory at this stage, the small parameter ϵ drops out of the equations and they can be solved for the first order perturbations aðrÞ and bðrÞ. Equation. (20) is an algebraic equation for bðrÞ that can be easily solved, yielding

ð−1Þp2p−2ðp − 1Þr2ðμ − 1Þ2p ðμ2 − 1ÞaðrÞ ra0ðrÞ κ2r2 bðrÞ¼α þ þ − p : ð22Þ μ2ðr2μÞp μ4 μ2 μ2 r

Inserting this result for bðrÞ in (19) and (21), while setting pr ¼ pl, we obtain one remaining partial differential equation, which can be solved for aðrÞ,

2 0 α2p−3ð4ðμ − 1Þμ2 þð5μ3 þ 7μ2 þ 3μ þ 1Þ 2 − ð9μ3 þ 3μ2 þ 3μ þ 1Þ Þ −3pð− ðμ−1Þ rÞp 2a p p r μ a00 þ − ¼ 0: ð23Þ r ðμ − 1Þμ2

In order to continue solving the equations, we will separate C2 ¼ 0. The Λ term appears due to a nonvanishing the study into two branches: (A) M ¼ 0, and pr ¼ pl ¼ cosmological constant we assumed as a first order matter −ρ ¼ −Λ as it was studied in Ref. [25], with the additional source. constraint p>1=2 to guarantee well-defined field equa- The solutions we find are completely different to the tions. We redo the calculations of [25], since we find a ones presented in [25]. The source of this discrepancy lies completely different result for the M ¼ 0.(B)M ≠ 0, in our choice of the tetrad (11), to which we associate a ρ ¼ pr ¼ pl ¼ 0 and p ¼ 2 to 10. For p ¼ 2 we reproduce vanishing spin connection. The tetrad chosen in the the result from [26]. previous work had the drawback that its torsion scalar, see Eq. (9) in [25], does not vanish in the Minkowski → 1 → 1 8 2 A. Minkowski background (M =0) spacetime limit A , B , but gives =r . In turn this leads to an infinite action for Minkowski spacetime. As we p>1=2 If one assumes , perturbations around a mentioned earlier, our tetrad avoids this complication by Minkowski background can be studied by setting M ¼ 0 μ ¼ 1 – having a vanishing torsion scalar for the Minkowski ( ). It can immediately be seen from Eqs. (19) (21) spacetime limit. that the influence of the teleparallel perturbation (e.g., the terms proportional to α) drops out. As a consequence one obtains the usual first order (A)dS Schwarzschild spacetime B. Schwarzschild background M ≠ 0 geometries as solutions of the perturbed field equations In this section, we will focus our study on perturbations with a cosmological constant as first order matter source, of Schwarzschild geometry (M ≠ 0) induced by weak ¼ ¼ −ρ ¼ −Λ ð Þ i.e., pr pl . The perturbation functions a r power-law fðTÞ gravity and bðrÞ are easily determined from (19)–(21). The metric coefficients AðrÞ and BðrÞ become 1 fðTÞ¼T þ αϵTp; ð26Þ 2 C1 1 2 2 AðrÞ¼1 þ ϵ C2 − − κ Λr ; ð24Þ r 3 exemplary for p ¼ 2 to p ¼ 4, since a solution for general p cannot be obtained. For higher integer values of p the C1 1 solutions have a similar form, but it is not further insightful BðrÞ¼1 þ ϵ þ κ2Λr2 ; ð25Þ r 3 to display the long expressions. The general structure of the vacuum solutions, i.e., ρ ¼ p ¼ p ¼ 0, for all p is for all p>1=2. Here C1 and C2 are integration constants r l labeling the linearized Schwarzschild solution of general 2M C1 relativity and a constant shift of the Minkowski metric AðrÞ¼1 − þ ϵ − þ C2 þ αa¯ðrÞ ; ð27Þ respectively. Usually they are chosen to be C1 ¼ 2M and r r

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1 ðC1 − 2C2MÞ large distance, limit. After the integration constants have BðrÞ¼ þ ϵ r r þ αb¯ðrÞ : ð28Þ been found, the solutions take the form 1 − 2M ð1 − 2MÞ2 r r 2M AðrÞ¼1 − þ ϵαaˆðrÞ; ð29Þ The integration constants C1 and C2 are determined in a r power series expansion in 1, such that the zeroth and first 1 r BðrÞ¼ þ ϵbˆðrÞ: ð30Þ order terms in this expansion vanish. Physically this means 1 − 2M we use the integration constants to avoid an influence of the r teleparallel perturbation in the weak field, respectively, For p ¼ 2 we reproduce the solutions found in [26]

2 2 þ 6 þ 2 16ð1 − 2MÞ3=2 ð1 − 3MÞ 2 M C1 M Mr r r r M AðrÞ¼1 − þ ϵ − þ C2 − α − þ ln 1 − ; ð31Þ r r Mr3 3M2 2M2 r

2 1 ðC1 − C2MÞ 8ð3M2 − 7Mr þ 2r2Þ 25M − 23r ln ð1 − 2MÞ BðrÞ¼ þ ϵ r r − α − þ þ r ; ð32Þ 1 − 2M ð1 − 2MÞ2 3 3ð1 − 2MÞ3=2 3ð1 − 2MÞ2 2 ð1 − 2MÞ2 r r Mr r r r Mr r which can be expressed conveniently in terms of the variable μ ¼ð1 − 2M=rÞ1=2, giving 2 3 4 6 2 2 C1 51 − 93μ − 128μ þ 45μ − 3μ − 12ð1 − 3μ Þ lnðμÞ AðrÞ¼μ þ ϵ − þ C2 − α ; ð33Þ r 6r2ð1 − μ2Þ2 C1 2 2 3 4 5 1 ð − ð1 − μ ÞC2Þ 63 − 24μ þ 12μ þ 64μ − 75μ þ 24μ − 12 lnðμÞ BðrÞ¼ þ ϵ r þ α : ð34Þ μ2 μ4 6r2μ4ð1 − μ2Þ

1 Determining the integration constants from the r expansions 16α 16α 1 1 16α 1 1 AðrÞ ∼ þ C2 − þ C1 þ O ;BðrÞ ∼ þ C1 − 2MC2 þ O ð35Þ 3M2 M r r2 3M r r2

2 yields C2 ¼ −16α=ð3M Þ and C1 ¼ −16α=M. For easy comparison with previous approaches we display the solutions (33) and (34) one more time with this choice of integration constants

13 − 99μ2 þ 128μ3 − 45μ4 þ 3μ6 þ 12ð1 − 3μ2Þ lnðμÞ AðrÞ¼μ2 þ ϵα ; ð36Þ 6r2ð1 − μ2Þ2

1 1 þ 24μ − 12μ2 − 64μ3 þ 75μ4 − 24μ5 þ 12 lnðμÞ BðrÞ¼ − ϵα : ð37Þ μ2 6r2μ4ð1 − μ2Þ

The leading order terms for the torsion scalar (12) for the weak squared power-law case in a Schwarzschild background behaves as

2ðμ − 1Þ2 13 − 36μ þ 108μ2 − 184μ3 þ 135μ4 − 36μ5 þ 12 lnðμÞ T ¼ − þ ϵα : ð38Þ μr2 6r4μ3

For the specific choice of C1 and C2 we find that in the M → 0 limit AðrÞ → 1, BðrÞ → 1, and T → 0. This result coincides with the one presented in [26]. For p ¼ 3 and p ¼ 4 we find similar solutions that can be expressed as

1 2ðμ − 1Þ2 AðrÞ¼μ2 þ ϵαaˆðrÞ;BðrÞ¼ þ ϵαbˆðrÞ;TðrÞ¼− þ ϵαTˆ ðrÞ: ð39Þ μ2 μr2

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(i) p ¼ 3

aˆðrÞ¼½−280μ12 þ 945μ11 þ 1120μ10 − 8295μ9 þ 6984μ8 þ 18060μ7 − 37632μ6 þ 1260μ5 þ 86520μ4 − 62909μ3 − 10080μ2 − 7560ð7μ2 − 1Þμ logðμÞþ178μ þ 2520ð315r4μðμ2 − 1Þ4Þ−1; ð40aÞ

bˆðrÞ¼½3780μ11 − 19040μ10 þ 27405μ9 þ 16560μ8 − 81480μ7 þ 56448μ6 þ 44100μ5 − 77280μ4 þ 23940μ3 þ 10080μ2 − 9553μ þ 7560μ logðμÞþ5040ð315r4μ5ðμ2 − 1Þ3Þ−1; ð40bÞ

Tˆ ðrÞ¼½7ðμ − 1Þð6300μ10 − 30380μ9 þ 44905μ8 þ 8545μ7 − 88055μ6 þ 74233μ5 þ 12493μ4 − 49667μ3 þ 27193μ2 − 10607μ − 2520Þþ7560μ logðμÞð315r6μ4ðμ2 − 1Þ2Þ−1: ð40cÞ

(ii) p ¼ 4

aˆðrÞ¼−4½−6435μ18 þ 36960μ17 − 32175μ16 − 221760μ15 þ 552552μ14 þ 145600μ13 − 1963962μ12 þ 1693120μ11 þ 2642640μ10 − 5436288μ9 þ 330330μ8 þ 7495488μ7 − 6846840μ6 − 4804800μ5 þ 3912986μ4 þ 2882880μ3 þ 55139μ2 þ 720720ð11μ2 − 1Þμ2 logðμÞ − 480480μ þ 45045ð15015r6μ2ðμ2 − 1Þ6Þ−1; ð41aÞ

bˆðrÞ¼−4½120120μ17 − 842985μ16 þ 1940400μ15 − 60060μ14 − 7141680μ13 þ 10198188μ12 þ 3443440μ11 − 20810790μ10 þ 13590720μ9 þ 12222210μ8 − 20612592μ7 þ 4504500μ6 þ 8888880μ5 − 6666660μ4 þ 720720μ3 þ 1451534μ2 − 720720μ2 logðμÞ − 1081080μ þ 135135ð15015r6μ6ðμ2 − 1Þ5Þ−1; ð41bÞ

Tˆ ðrÞ¼2½1441440μ2 logðμÞ − 2ðμ − 1Þð210210μ16 − 1443585μ15 þ 3379695μ14 − 1004685μ13 − 9227445μ12 þ 15024783μ11 − 270497μ10 − 21081287μ9 þ 18275173μ8 þ 4731643μ7 − 15880949μ6 þ 7362271μ5 þ 2197111μ4 − 3388469μ3 þ 1656571μ2 þ 225225μ − 45045Þð15015r8μ5ðμ2 − 1Þ4Þ−1: ð41cÞ

For the other higher values of p the calculation follows the d ∂L ∂L same scheme and their solutions are similar to the above − ¼ 0; ð42Þ dτ ∂q_ μ ∂qμ ones. Since they are lengthy and not insightful, we will not present them explicitly. of the Lagrangian

IV. PARTICLE MOTION PHENOMENOLOGY μ ν 2L ¼ gμνq_ q_ To relate the influence of a teleparallel modification of 2M 1 general relativity to observables, we study the motion of ¼ 1 − þ ϵaðrÞ t_2 − þ ϵbðrÞ r_2 r 1 − 2M test particles in the background solution defined by the r 2 2 metric coefficients (31) and (32). We explicitly derive the − r2θ_ − r2 sin2 θϕ_ ; ð43Þ photon sphere around the black hole and the perihelion shift of nearly circular orbits. Nowadays the photon sphere where qμðτÞ¼ðtðτÞ;xðτÞ; θðτÞ; ϕðτÞÞ, and q_ μ denotes the is of particular interest since it defines the edge of the μ shadow of a black hole while the perihelion shift was derivative of q with respect to the affine parameter τ. The already derived in [26], but on the basis of an erroneous perturbation functions aðrÞ and bðrÞ under consideration solution, as we discussed above. can be read off in (31) and (32). To solve the EL equations we employ the usual scheme for spherically symmetric spacetimes: we restrict ourselves A. Geodesic equation and effective potential to motion in the equatorial plane and set θ ¼ π=2, and we The worldline qðτÞ of test particles in a curved spacetime derive the usual conserved quantities the energy k and is determined by the Euler-Lagrange (EL) equations angular momentum h

084064-7 BAHAMONDE, FLATHMANN, and PFEIFER PHYS. REV. D 100, 084064 (2019) pffiffiffi ∂L 2M ¼ 3 ¼3 3 ð Þ k ¼ ¼ 1 − þ ϵaðrÞ t;_ ð44Þ r0 M; h0 k0M 51 ∂t_ r and the first order perturbation gives, for the different ∂L 2 values of p, the following numerical values: h ¼ ¼ r ϕ_ : ð45Þ ∂ϕ_ −6 α ðp ¼ 2Þ r1 ≈ 14133.8000 × 10 ; We obtain the effective potential to first order in ϵ from the M α constancy of the Lagrangian, expressed in terms of the ð ¼ 3Þ ≈−1362 5400 10−6 ð Þ p r1 . × 3 ; 52 conserved quantities M α 2 2 −6 aðrÞ k 1 h ðp ¼ 4Þ r1 ≈ 121.3220 × 10 ; 1 − ϵ − þ ϵbðrÞ r_2 − M5 1 − 2M 1 − 2M 1 − 2M r2 r r r −6 α ðp ¼ 5Þ r1 ≈−10.2582 × 10 7 ; ð53Þ þ Oðϵ2Þ¼σ; ð46Þ M

−6 α where σ ¼ 0 for massless particles and σ ¼ 1 for massive ðp ¼ 6Þ r1 ≈ 8.3757 × 10 ; M9 particles. For further calculations we rearrange Eq. (46) as α ð ¼ 7Þ ≈−0 6670 10−6 ð Þ p r1 . × 11 ; 54 1 1 1 2 2 1 2 M 0 ¼ _2 − 2 þ h 1 − M þ σ 1 − M 2 r 2 k 2 2 2 α r r r ð ¼ 8Þ ≈ 0 0521 10−6 p r1 . × 13 ; ϵ aðrÞ 2M M þ 2 þ ð Þ 1 − α k 2M b r −6 2 1 − r ðp ¼ 9Þ r1 ≈−0.0040 × 10 ; ð55Þ r M15 2 2 2 2 2 − ð Þ h 1 − M − σ ð Þ 1 − M ð Þ b r 2 b r ; 47 −6 α r r r ðp ¼ 10Þ r1 ≈ 0.0003 × 10 : ð56Þ M17 so we can read off the effective potential to first order in ϵ We clearly see that for positive α and even p the 2 teleparallel perturbation of general relativity yields a larger 1 1 2M h VðrÞ¼− k2 þ 1 − þ σ photon sphere around a spherically symmetric black hole 2 2 r r2 and thus predicts a larger black hole shadow. For odd p a ϵ aðrÞ 2M smaller shadow is predicted. Moreover it is evident that the þ ½k2 þ bðrÞ 1 − 2 1 − 2M r larger p the smaller the first order influence of the tele- r 2 2 2 parallel perturbation. The relation between the photon − ð Þ σ þ h 1 − M ð Þ b r 2 48 sphere and teleparallel perturbations of general relativity r r is investigated here for the first time. For circular timelike orbits, σ ¼ 1, it is also possible to from solve the equations V ¼ 0 and V0 ¼ 0. However the 1 appearing expressions are not very insightful. The impor- _2 þ ð Þ¼0 ð Þ tant observation is that teleparallely perturbed general 2 r V r : 49 relativity, not surprisingly, allows for circular orbits, which For the analysis of the perihelion shift, it is necessary to will be the starting point for the derivation of the perihelion reparametrize rðτÞ as rðϕÞ, which amounts to the equation shift now. We consider a perturbation around a circular ðϕÞ¼ þ ðϕÞ orbit rc and plug the ansatz r rc rϕ into (50) 1 r_2 1 1 dr 2 r4 and obtain þ VðrÞ¼ þ VðrÞ¼0: ð50Þ 2 ϕ_ 2 ϕ_ 2 2 ϕ 2 d h 2 4 drϕ ðr þ rϕÞ ¼ −2 c ð þ Þ ð Þ 2 V rc rϕ : 57 B. Photon sphere and perihelion shift dϕ h For circular orbits (e.g., r ¼ const; r_ ¼ 0) the effective Assuming that the ratio rϕ=rc is small, the right-hand side potential and its derivative have to vanish. We perturb the can be expanded into powers of this parameter to second r ¼ r0 þ ϵr1 radial coordinate of the circular orbit c , the order angular momentum h ¼ h0 þ ϵh1, and the energy k ¼ k0 þ ϵ ¼ 0 0 ¼ 0 k1 and solve both equations V and V order by 2 4 3 drϕ r rϕ order. For circular photon orbits, σ ¼ 0, solving the zeroth ¼ − c 00ð Þ 2 þ O ð Þ ϕ 2 V rc rϕ 3 ; 58 order equations yields d h r0

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ð Þ¼0 8 2 48 4 where we used that for circular orbits V rc and ˆ q ˆ q 0 Δϕ ¼2 ¼ ; Δϕ ¼3 ¼ − ; V ðr Þ¼0, as discussed above. The solution rϕ thus p 2 p 4 c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rc rc 4 rc 00 6 8 oscillates with the wave number K ¼ 2 V ðr Þ and 192q 640q h c Δˆ ϕ ¼ Δˆ ϕ ¼ − p¼4 6 ; p¼5 8 ; the perihelion shift is given by rc rc 1920 10 5376 12 Δˆ ϕ ¼ q Δˆ ϕ ¼ − q 1 h p¼6 10 p¼7 12 ; Δϕ ¼ 2π − 1 ¼ 2π pffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 1 : ð59Þ rc rc K r2 V00ðr Þ 14336 14 36864 16 c c Δˆ ϕ ¼ q Δˆ ϕ ¼ − q p¼8 14 ; p¼9 16 ; rc rc To derive the explicit expression for the perihelion shift for 92160 18 σ ¼ 1 Δˆ ϕ ¼ q ð Þ massive objects we consider the potential V with , see p¼10 18 : 65 (48), its first derivative V0 and its second derivative V00.We rc evaluate the equations Vðr Þ¼0 and V0ðr Þ¼0 with c c The qualitative behavior of the perihelion shift is always the h ¼ h0 þ ϵh1 and k ¼ k0 þ ϵk1. The zeroth order of these same, only the numerical values differ. As for the photon equations determines h0ðr Þ and k0ðr Þ as c c sphere, the higher p, the smaller the influence of the pffiffiffiffiffi teleparallel perturbation and corrections to the perihelion 2 − ¼ Mrc ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM rc ð Þ shift appear only in higher orders in q. h0 − 3 ;k0 : 60 rc M rcðrc − 3MÞ Since the influence of the teleparallel perturbation decreases with higher power in p the most strict bound on α is obtained for p ¼ 2 and is the one obtained in [26]. The first order determines h1ðr Þ and k1ðr Þ. Depending on c c We expect to be able to find stronger bounds from the the choice of the sign of h0 we obtain two different upcoming study on teleparallel perturbations of rotating solutions for h1 (the sign of k0 is irrelevant here) black holes. r2ð2Maðr Þ − r ðr − 2MÞa0ðr ÞÞ ¼∓ c pc ffiffiffiffiffipcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic c ð Þ h1 3 : 61 V. CONCLUSION 4 M rc − 3M In this paper we presented the first order influence of a ð Þ The sign labeling h1 refers to the sign chosen of the zeroth teleparallel power-law f T gravity perturbation of general order h0, which was chosen to calculate h1. There is no relativity, in spherical symmetry. The central results of this need to derive k1 explicitly, since it does not enter the article are as follows: perihelion shift equation. Having obtained the constants of (i) To first order, a power-law perturbation of the type ð Þ¼ þ α p motion for the circular orbit we can derive the perihelion f T T 2 T yields no teleparallel correction to 00 1 2 shift by plugging the values into V ðr; k0;h0;h1Þ to obtain Minkowski spacetime for p> = . 00 (ii) The explicit derivation of the first order teleparallel V ðrcÞ alone. Due to the different solutions for the ð Þ¼ þ α p constants of motion there exist two options to derive the f T T 2 T corrections to Schwarzschild perihelion shift geometry for p ¼ 2 to p ¼ 10, displayed in Eqs. (33) and (34) for p ¼ 2 and in (40) and (41) for p ¼ 3 and p ¼ 4, respectively. The perturbed solutions for Δϕðh0þ;h1þÞ; Δϕðh0−;h1−Þ; ð62Þ higher power-law parameter p have the same structure but they are lengthy and for this reason, we do which are related to each other through not present them. The latter allowed us to calculate the teleparallel modifica- Δϕðh0−;h1−Þ¼−4π − Δϕðh0þ;h1þÞ: ð63Þ tions of the photon sphere and the perihelion shift: two observables which are experimentally accessible and can be used to check the viability of fðTÞ models. For both Expanding the perihelion shift into a power series in the observables we find that the larger p, the smaller the influence variable q ¼ M yields rc of the teleparallel modification. Thus, the p ¼ 2 model is most constraint from the perihelion shift of mercury, which is Δϕð Þ¼6π þ 27π 2 þ Oð 3ÞþϵπΔˆ ϕ þ Oðϵ2Þ jαj ¼ 2 20 1020 2 h0þ;h1þ q q q p : max . × km according to [26].Weexpectto find further, stronger constraints, for the different models by ð64Þ studying rotating black holes and their phenomenology. The results we presented improve and extend existing ˆ Δϕp is the leading order teleparallel perturbation of the results on first order power-law fðTÞ models, which were usual GR result. For the different values of p it is given by presented in [25,26]. In the first article the tetrad chosen

084064-9 BAHAMONDE, FLATHMANN, and PFEIFER PHYS. REV. D 100, 084064 (2019) was not compatible with a vanishing spin connection and weak power-law fðTÞ-gravity to systematically check its the field equations were incorrect. During our derivations, consistency with observations. The next step in this we paid particular attention to present all necessary steps in program is to consider axially symmetric perturbations the perturbation theory, so that our results are easily around Kerr spacetime, to derive the change in the photon reproducible. regions, which will have an imprint on the predictions of An important opportunity that our presented approach the shape of the black hole’s shadow. here offers is to investigate the connection between the vanishing of the first order contributions of teleparallel ACKNOWLEDGMENTS corrections around Minkowski spacetime and the nonvanishing of the corrections around Schwarzschild geometry, The authors would like to thank Jackson Levi Said, with the degrees of freedom of fðTÞ gravity; the latter being Gabriel Farrugia, Andrew DeBenedictis, and Sasa Ilijic for debated in the literature [32–34]. Here we considered static very fruitful discussions. C. P. was supported by the perturbations; in the future, nonstatic spherically and axially Estonian Research Council and the European Regional symmetric gravitational waves from weak power-law fðTÞ Development Fund through the Center of Excellence gravity will be investigated and complement the gravita- TK133 “The Dark Side of the Universe.” K. F. gratefully tional wave analysis of fðTÞ gravity around Minkowski and acknowledges support by the DFG, within the Research FLRW geometry [32,35,36] and also at the astrophysical Training Group Models of Gravity and mobility funding level with compact binary coalescences [37]. from the European Regional Development Fund through This paper is a first step towards a complete phenom- Dora Plus. S. B. is supported by Mobilitas Pluss N° enological catalog of observables, which shall be derived in MOBJD423 by the Estonian government.

[1] B. P. Abbott et al. (LIGO Scientific and Virgo Collabora- New Schwarzschild-like solutions in fðTÞ gravity through tions), Observation of Gravitational Waves from a Binary Noether symmetries, Phys. Rev. D 89, 104042 (2014). Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016). [13] M. H. Daouda, M. E. Rodrigues, and M. J. S. Houndjo, [2] K. Akiyama et al. (Event Horizon Telescope Collaboration), Anisotropic fluid for a set of non-diagonal tetrads in First M87 Event Horizon Telescope results. I. The shadow fðTÞ gravity, Phys. Lett. B 715, 241 (2012). of the supermassive black hole, Astrophys. J. 875,L1 [14] J. Aftergood and A. DeBenedictis, Matter conditions for (2019). regular black holes in fðTÞ gravity, Phys. Rev. D 90, 124006 [3] R. Aldrovandi and J. G. Pereira, Teleparallel Gravity (2014). (Springer, Dordrecht, 2013), Vol. 173. [15] C. G. Böhmer and F. Fiorini, The regular black hole in four [4] Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. dimensional Born–Infeld gravity, Classical Quantum Grav- Saridakis, fðTÞ teleparallel gravity and cosmology, Rep. ity 36, 12LT01 (2019). Prog. Phys. 79, 106901 (2016). [16] C. G. Boehmer, T. Harko, and F. S. N. Lobo, Wormhole [5] R. Ferraro and F. Fiorini, Modified teleparallel gravity: geometries in modified teleparralel gravity and the energy Inflation without inflaton, Phys. Rev. D 75, 084031 (2007). conditions, Phys. Rev. D 85, 044033 (2012). [6] G. R. Bengochea and R. Ferraro, Dark torsion as the cosmic [17] M. Jamil, D. Momeni, and R. Myrzakulov, Wormholes in a speed-up, Phys. Rev. D 79, 124019 (2009). viable fðTÞ gravity, Eur. Phys. J. C 73, 2267 (2013). [7] Y.-F. Cai, S.-H. Chen, J. B. Dent, S. Dutta, and E. N. [18] S. Bahamonde, U. Camci, S. Capozziello, and M. Jamil, Saridakis, Matter bounce cosmology with the f(T) gravity, Scalar-tensor Teleparallel wormholes by noether sym- Classical Quantum Gravity 28, 215011 (2011). metries, Phys. Rev. D 94, 084042 (2016). [8] K. Bamba, C.-Q. Geng, C.-C. Lee, and L.-W. Luo, Equation [19] A. Jawad and S. Rani, Lorentz distributed noncommutative of state for dark energy in fðTÞ gravity, J. Cosmol. wormhole solutions in extended teleparallel gravity, Eur. Astropart. Phys. 01 (2011) 021. Phys. J. C 75, 173 (2015). [9] J. B. Dent, S. Dutta, and E. N. Saridakis, fðTÞ gravity [20] D. Horvat, S. Ilijić, A. Kirin, and Z. Narančić, Nonmini- mimicking dynamical dark energy. Background and pertur- mally coupled scalar field in teleparallel gravity: Boson bation analysis, J. Cosmol. Astropart. Phys. 01 (2011) 009. stars, Classical Quantum Gravity 32, 035023 (2015). [10] N. Tamanini and C. G. Boehmer, Good and bad tetrads in [21] C. G. Boehmer, A. Mussa, and N. Tamanini, Existence of fðTÞ gravity, Phys. Rev. D 86, 044009 (2012). relativistic stars in fðTÞ gravity, Classical Quantum Gravity [11] R. Ferraro and F. Fiorini, Spherically symmetric static 28, 245020 (2011). spacetimes in vacuum fðTÞ gravity, Phys. Rev. D 84, [22] S. Ilijic and M. Sossich, Compact stars in fðTÞ extended 083518 (2011). theory of gravity, Phys. Rev. D 98, 064047 (2018). [12] A. Paliathanasis, S. Basilakos, E. N. Saridakis, S. [23] M. Pace and J. L. Said, A perturbative approach to neutron Capozziello, K. Atazadeh, F. Darabi, and M. Tsamparlis, stars in fðT;T Þ gravity, Eur. Phys. J. C 77, 283 (2017).

084064-10 PHOTON SPHERE AND PERIHELION SHIFT IN WEAK FðTÞ … PHYS. REV. D 100, 084064 (2019)

[24] M. Pace and J. L. Said, Quark stars in fðT;T Þ-gravity, Eur. [31] M. Hohmann, L. Järv, M. Krššák, and C. Pfeifer, Modified Phys. J. C 77, 62 (2017). teleparallel theories of gravity in symmetric spacetimes, [25] M. L. Ruggiero and N. Radicella, Weak-field spherically Phys. Rev. D 100, 084002 (2019). symmetric solutions in fðTÞ gravity, Phys. Rev. D 91, [32] A. Golovnev and T. Koivisto, Cosmological perturbations in 104014 (2015). modified teleparallel gravity models, J. Cosmol. Astropart. [26] A. DeBenedictis and S. Ilijic, Spherically symmetric vac- Phys. 11 (2018) 012. α 2 γ uum in covariant FðTÞ¼T þ 2 T þ OðT Þ gravity theory, [33] R. Ferraro and M. J. Guzmán, Quest for the extra degree of Phys. Rev. D 94, 124025 (2016). freedom in fðTÞ gravity, Phys. Rev. D 98, 124037 (2018). [27] M. Krššák and E. N. Saridakis, The covariant formulation of [34] R. Ferraro and M. J. Guzmán, Hamiltonian formalism for fðTÞ gravity, Classical Quantum Gravity 33, 115009 fðTÞ gravity, Phys. Rev. D 97, 104028 (2018). (2016). [35] G. Farrugia, J. L. Said, V. Gakis, and E. N. Saridakis, [28] A. Golovnev, T. Koivisto, and M. Sandstad, On the Gravitational waves in modified teleparallel theories, Phys. covariance of teleparallel gravity theories, Classical Quan- Rev. D 97, 124064 (2018). tum Gravity 34, 145013 (2017). [36] R. C. Nunes, S. Pan, and E. N. Saridakis, New observational [29] M. Hohmann, L. Järv, M. Krššák, and C. Pfeifer, Tele- constraints on fðTÞ gravity through gravitational-wave parallel theories of gravity as analogue of nonlinear electro- astronomy, Phys. Rev. D 98, 104055 (2018). dynamics, Phys. Rev. D 97, 104042 (2018). [37] R. C. Nunes, M. E. S. Alves, and J. C. N. de Araujo, Fore- [30] M. Hohmann, L. Järv, and U. Ualikhanova, Covariant cast constraints on fðTÞ gravity with gravitational waves formulation of scalar-torsion gravity, Phys. Rev. D 97, from compact binary coalescences, Phys. Rev. D 100, 104011 (2018). 064012 (2019).

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