In the present paper, we consider the possibility of observationally constraining Hořava gravity at the scale of the Solar System, by considering the classical tests of general relativity (perihelion precession of the planet Mercury, deflection of light by the Sun and the radar echo delay) for the spherically symmetric black hole Kehagias–Sfetsos solution of Hořava–Lifshitz gravity. All these gravitational effects can be fully explained in the framework of the vacuum solution of Hořava gravity. Moreover, the study of the classical general relativistic tests also constrains the free parameter of the solution. From the analysis of the perihelion precession of the planet Mercury, we obtain for the free parameter ω of the Kehagias–Sfetsos solution the constraint ω≥3.212×10−26 cm−2, the deflection of light by the Sun gives ω≥4.589×10−26 cm−2, while the radar echo delay observations can be explained if the value of ω satisfies the constraint ω≥9.179×10−26 cm−2.
Recently, a renormalizable gravity theory in four dimensions that reduces to Einstein gravity with a non-vanishing cosmological constant in the infrared (IR) energy scale (corresponding to large distances), but with improved ultraviolet (UV) energy-scale behaviours (corresponding to very small distances), was proposed by Hořava (2009b,c). Quantum field theory has had considerable success experimentally, but, from a theoretical point of view, it predicts infinite values for physical quantities. Thus, for certain Feynman diagrams containing loops, the calculations lead to an infinite result. This is known as the UV divergence of quantum field theory, because small loop sizes correspond to high energies. IR divergence is the divergence caused by the low-energy behaviour of a quantum theory. The Hořava theory admits a Lifshitz scale invariance in time and space, exhibiting a broken Lorentz symmetry at short scales, while, at large distances, higher derivative terms do not contribute, and the theory reduces to standard general relativity (GR). The Hořava theory has received a great deal of attention and since its formulation various properties and characteristics have been extensively analysed, ranging from formal developments (Appignani et al. 1999; Afshordi 2009; Blas et al. 2009; Cai et al. 2009a,b; Calcagni 2009a; Charmousis et al. 2009; Germani et al. 2009; Hořava 2009a; Iengo et al. 2009; Kluson 2009; Kobakhidze 2009; Kocharyan 2009; Li & Pang 2009; Mukohyama 2009a; Nishioka 2009; Orlando & Reffert 2009; Sotiriou et al. 2009a,b; Visser 2009; Chen & Huang 2010; Kluson 2010), cosmology (Boehmer & Lobo 2009; Brandenberger 2009; Cai & Saridakis 2009; Calcagni 2009b; Chen et al. 2009; Gao 2009; Gao et al. 2009; Ha et al. 2009; Kalyana Rama 2009; Kiritsis & Kofinas 2009; Kobayashi et al. 2009; Minamitsuji 2009; Mukohyama 2009b; Mukohyama et al. 2009; Nojiri & Odintsov 2009; Piao 2009; Takahashi & Soda 2009; Wang & Maartens 2009; Wang & Wu 2009), dark energy (Park 2009a; Saridakis 2009) and dark matter (Mukohyama 2009c), spherically symmetric or rotating solutions (Botta-Cantcheff et al. 2009; Cai et al. 2009c,d; Castillo & Larranaga 2009; Colgain & Yavartanoo 2009; Kehagias & Sfetsos 2009; Kim et al. 2009; Lee et al. 2009; Lu et al. 2009; Mann 2009; Myung & Kim 2009; Nastase 2009; Park 2009b; Peng & Wu 2009; Chen & Jing 2010; Ghodsi & Hatefi 2010), to the weak field observational tests (Iorio & Ruggiero in press a,b). At large distances, higher derivative terms do not contribute and the theory reduces to standard GR if a particular coupling λ, which controls the contribution of the trace of the extrinsic curvature, has a specific value. Indeed, λ is running, and if λ=1 is an IR fixed point, standard GR is recovered. Therefore, although a generic vacuum of the theory is the anti-de Sitter one, particular limits of the theory allow for the Minkowski vacuum, a physical state characterized by the absence of ordinary (baryonic) matter. In this limit, post-Newtonian coefficients coincide with those of pure GR. Thus, the deviations from conventional GR can be tested only beyond the post-Newtonian corrections, that is, for a system with strong gravity at astrophysical scales.
In this context, IR-modified Hořava gravity seems to be consistent with the current observational data, but, in order to test its viability, more observational constraints are necessary. In Konoplya (2009), potentially observable properties of black holes in the Hořava–Lifshitz gravity with Minkowski vacuum were considered, namely the gravitational lensing and quasi-normal modes. Quasi-normal modes are the modes of energy dissipation of a perturbed object or field. Black holes have many quasi-normal modes that describe the exponential decrease in asymmetry of the black hole in time as it evolves towards the spherical shape. It was shown that the bending angle is seemingly smaller in the considered Hořava–Lifshitz gravity than in GR, and the quasi-normal modes of black holes are longer lived, and have larger real oscillation frequency in Hořava–Lifshitz gravity than in GR. In Chen & Jing (2009), by adopting the strong field limit approach, the properties of strong field gravitational lensing for the Hǒrava–Lifshitz black hole were considered, and the angular position and magnification of the relativistic images were obtained. Compared with the Reissner–Norström black hole, a significant difference in the parameters was found. Thus, it was argued that this may offer a way to distinguish a deformed Hǒrava–Lifshitz black hole from a Reissner–Nordström black hole. In GR, the Reissner–Nordström black hole solution describes the gravitational field of a charged black hole. In Chen & Wang (2010), the behaviour of the effective potential was analysed, and the time-like geodesic motion in the Hořava–Lifshitz space–time was also explored.
In Harko et al. (2009), the basic physical properties of matter forming a thin accretion disc in the vacuum space–time metric of the Hořava–Lifshitz gravity models were considered. It was shown that significant differences from the general relativistic case exist, and that the determination of these observational quantities could discriminate, at least in principle, between standard GR and Hořava–Lifshitz theory, and constrain the parameter of the model.
It is the purpose of the present paper to consider the classical tests (perihelion precession, light bending and the radar echo delay) of GR for static gravitational fields in the framework of Hořava–Lifshitz gravity. To do this, we shall adopt for the geometry outside a compact, stellar-type object (the Sun), the static and spherically symmetric metric obtained by Kehagias & Sfetsos (2009). For the Kehagias and Sfetsos (KS) metric, we first consider the motion of a particle (planet), and analyse the perihelion precession. In addition to this, by considering the motion of a photon in the static KS field, we study the bending of light by massive astrophysical objects and the radar echo delay. All these gravitational effects can be explained in the framework of the KS geometry. Existing data on light-bending around the Sun, using long-baseline radio interferometry, ranging to Mars using the Viking lander, the perihelion precession of Mercury and recent lunar laser ranging results can all give significant and detectable Solar System constraints, associated with Hořava–Lifshitz gravity. More exactly, the study of the classical general relativistic tests constrains the parameter of the solution. We will also compare our results with the phenomenological constraints on the KS solution from Solar System orbital motions obtained in Iorio & Ruggiero (in press a). In order to obtain reliable constraints on the parameter of the model, the analysis must not be limited to the perihelion precession of the planet Mercury, but it should be extended to take into account the other recent observational data on the perihelion precession of the other planets of the Solar System.
In the context of the classical tests of GR, the Dadhich, Maartens, Papadopoulos and Rezania (DMPR) solution of the spherically symmetric static vacuum field equations in brane world models was also extensively analysed (Boehmer et al. 2008, 2010). It was found that the existing observational Solar System data constrain the numerical values of the bulk tidal parameter and of the brane tension.
This paper is organized as follows. In §2, we present the action and specific solutions of static and spherically symmetric space–times in Hořava–Lifshitz gravity. In §3, we analyse the classical Solar System tests for the case of the KS asymptotically flat solution (Kehagias & Sfetsos 2009) of Hořava–Lifshitz gravity. We conclude our results in §4.
2. Black holes in Hořava–Lifshitz gravity
In the Hořava–Lifshitz gravity theory, the Lorentz symmetry is broken in the UV. The breaking manifests in the strong anisotropic scalings between space and time, , . In (3+1)-dimensional space–times, the theory is powercounting renormalizable, provided that z≥3. At low energies, the theory is expected to flow to z=1, whereby the Lorentz invariance is accidentally restored. Such an anisotropy between time and space can be easily realized, when writing the metric in the Arnowitt–Deser–Misner (ADM) form (Misner et al. 1973). The formalism supposes that the space–time is foliated into a family of space-like surfaces Σt, labelled by their time coordinate t, and with coordinates on each slice given by xi. Using the ADM formalism, the four-dimensional metric is parametrized in the following form: 2.1 The Einstein–Hilbert action is given by 2.2 where G is Newton’s constant, R(3) is the three-dimensional curvature scalar for gij and Kij is the extrinsic curvature, defined as 2.3 where the dot denotes a derivative with respect to t.
The IR-modified Hořava action is given by (Park 2009b) 2.4 where κ, λg, νg, μ, ω and ΛW are constant parameters; is the contraction of the intrinsic curvature, while ∇j is the covariant derivative with respect to the three-dimensional metric; Cij is the Cotton tensor, defined as 2.5 In the UV, this theory is power-counting renormalizable, at least around the flat space (vacuum) solution. In the IR, the terms of the lowest dimension should dominate. Note that the last term in equation (2.4) represents a ‘soft’ violation of the ‘detailed balance’ condition, which modifies the IR behaviour. More specifically, if one maintains Hořava’s original detailed balance, and one tries to recover Einstein–Hilbert in the low-energy regime, then one obtains a negative cosmological constant. The introduction of the last term in equation (2.4) corrects this, and provides a positive cosmological constant. Since ω∝μ2, this IR modification term, μ4R(3), with an arbitrary cosmological constant, represents the analogues of the standard Schwarzschild-(A)dS solutions, which were absent in the original Hořava model.
The fundamental constants of the speed of light c, Newton’s constant G and the cosmological constant Λ are defined as 2.6
The Hořava–Lifshitz theory is associated with the breaking of the diffeomorphism invariance, required for the anisotropic scaling in the UV (Hořava 2009c).
Throughout this work, we consider the static and spherically symmetric metric given by 2.7 where eν(r) and eλ(r) are arbitrary functions of the radial coordinate r.
Imposing the specific case of λg=1, which reduces to the Einstein–Hilbert action in the IR limit, one obtains the following solution of the vacuum field equations in the Hořava gravity: 2.8 where β is an integration constant (Park 2009b).
By considering β=−α2/ΛW and ω=0, the solution given by equation (2.8) reduces to the Lu, Mei and Pope (LMP) solution (Lu et al. 2009), given by 2.9 Alternatively, considering now β=4ωM and ΛW=0, one obtains the KS asymptotically flat solution (Kehagias & Sfetsos 2009), given by 2.10 If the limit 4M/ωr3≪1, from equation (2.10), we obtain the standard Schwarzschild metric of GR, eν(r)=1−2M/r, which represents a ‘post-Newtonian’ approximation of the KS solution of the second order in the speed of light. We shall use the KS solution for analysing the Solar System constraints of the theory. Note that there are two event horizons at 2.11 To avoid a naked singularity at the origin, one also needs to impose the condition 2.12 Note that in the GR regime, i.e. ωM2≫1, the outer horizon approaches the Schwarzschild horizon, r+≃2M, and the inner horizon approaches the central singularity, r−≃0. One should also note that the KS solution is obtained without requiring the projectability condition, which was assumed in the original Hořava theory. However, it is important to emphasize that static and spherically symmetric exhibiting the projectability condition have been obtained in the literature (Greenwald et al. 2010; Tang & Chen 2010; Wang et al. 2010).
3. Solar System tests for Hořava–Lifshitz gravity black holes
At the level of the Solar System, there are three fundamental tests, which can provide important observational evidence for GR and its generalizations, and for alternative theories of gravitation in flat space. These tests are the precession of the perihelion of Mercury, the deflection of light by the Sun and the radar echo delay observations and they have been used to successfully test the Schwarzschild solution of GR and some of its generalizations. In this section, we consider these standard Solar System tests of GR in the case of the KS asymptotically flat solution (Kehagias & Sfetsos 2009) of Hořava–Lifshitz gravity. Throughout the next section, we use the natural system of units with G=c=1.
(a) The perihelion precession of the planet Mercury
The motion of a test particle in the gravitational field of the metric given by equation (2.7) can be derived from the variational principle 3.1 where the dot denotes d/ds. It may be verified that the orbit is planar, and hence without any loss of generality we can set θ=π/2. Therefore, we will use ϕ as the angular coordinate. Since neither t nor ϕ appears explicitly in equation (3.1), their conjugate momenta are constant, 3.2
The change of variable r=1/u and the substitution d/ds=Lu2 d/dϕ transform equation (3.3) into the form 3.4 By formally representing e−λ=1−f(u), we obtain 3.5 By taking the derivative of the previous equation with respect to ϕ, we find 3.6 where 3.7 and we have denoted
A circular orbit u=u0 is given by the root of the equation u0=F(u0). Any deviation δ=u−u0 from a circular orbit must satisfy the equation 3.8 which is obtained by substituting u=u0+δ into equation (3.6). Therefore, in the first order in δ, the trajectory is given by 3.9 where δ0 and β are constants of integration. The angles of the perihelia of the orbit are the angles for which r is minimum, and hence u or δ is maximum. Therefore, the variation of the orbital angle from one perihelion to the next is 3.10
The quantity σ defined by the above equation is called the perihelion advance, which represents the rate of advance of the perihelion. As the planet advances through ϕ radians in its orbit, its perihelion advances through σϕ radians. From equation (3.10), σ is given by 3.11 or, for small (dF/du)u=u0, by 3.12
For a complete rotation, we have ϕ≈2π(1+σ), and the advance of the perihelion is 3.13
The observed value of the perihelion precession of the planet Mercury is δϕobs=43.11±0.21 arcsec per century (Shapiro et al. 1971, 1976). As a first step in the study of the perihelion precession in Hořava–Lifshitz gravity, the relevant functions are given by 3.14
The circular orbits are given by the roots of the equation F(u0)=u0, which is given by 3.15
In order to solve equation (3.15), we represent the parameter ω as ω=ω0/M2, and u0 as u0=x0/M, where ω0 and x0 are dimensionless parameters. Then, equation (3.15) can be written as 3.16 where b2=M2/L2.
In the case of the planet Mercury, we have a=57.91×1011 cm and e=0.205615, while for the values of the mass of the Sun and of the physical constants, we take M=M⊙=1.989×1033 g,c=2.998×1010 cm s−1 and G=6.67×10−8 cm3g−1s−2 (Anderson et al. 1987). Mercury also completes 415.2 revolutions each century. With the use of these numerical values we first obtain b2=M/a(1−e2)=2.66136×10−8. By performing a first-order series expansion of the square root in equation (3.16), we obtain the standard general relativistic equation , with the physical solution . In the general case, the value of x0 also depends on the numerical value of ω0, and, for a given ω0, x0 must be obtained by numerically solving the nonlinear algebraic equation (3.16). The variation of x0 as a function of ω0 is represented in figure 1.
The perihelion precession is given by δϕ=π(dF(u)/du)|u=u0, and, in the variables, x0 and ω0 can be written as 3.17 where 3.18
In the ‘post-Newtonian’ limit , we obtain the classical general relativistic result δϕGR=6πb2. The variation of the perihelion precession angle as a function of ω0 is represented in figure 2.
The gravity analysis of radio Doppler and range data generated by the Deep Space Network with Mariner 10 during two of its encounters with Mercury in March 1974 and March 1975 determined the observed value of the perihelion precession of the planet Mercury as δϕobs=43.11±0.20 arcsec per century (Anderson et al. 1987). Therefore, the range of variation of the perihelion precession is δϕobs∈(42.91,43.31). This range of observational values fixes the range of ω0 as ω0∈(6.95508×10−16,6.98748×10−16). The general relativistic formula for the precession gives δϕGR=42.94 arcsec per century. The perihelion precession can also be obtained by using elliptic integrals, or, in the case of the arbitrary central potentials, by using the method developed in Schmidt (2008).
Recently, new observational results on the extra-precession of the perihelion of the planets of the Solar System have been obtained. These results can be used as a test for gravitational models in the Solar System, such as the determinations of the PPN parameter β and of the coefficient J2 of the oblateness of the Sun (Pitjeva 2005, 2009; Fienga et al. 2009; Laskar et al. 2009). One can also use these results to estimate a possible advance in the planets perihelia, and an anomalous precession to the usual Newtonian/Einsteinian secular precession of the longitude of the perihelion of Saturn was found (Iorio 2009). Once the precision of the observations improves, these data could also be used to constrain through perihelion precession the value of the parameter ω for the KS black hole solution.
(b) Light deflection by the Sun
The deflection angle of light rays passing nearby the Sun in the KS geometry is given, with the use of the general relation derived in Weinberg (1972), by 3.19 where r0 is the distance of the closest approach, and we have denoted 3.20 By introducing a new variable x by means of the transformation r=r0x, equation (3.19) can be written as 3.21 By representing ω as ω=ω0/M2, and r0 as r0=x0M, we obtain 3.22 where we have denoted 3.23 and 3.24
For the Sun, by taking r0=R⊙=6.955×1010 cm, where R⊙ is the radius of the Sun, we find for x0 the value x0=4.71194×105. The variation of the deflection angle is represented, as a function of ω0, in figure 3. In the ‘post-Newtonian’ limit , we obtain the classical general relativistic result Δϕ=4M⊙/R⊙=1.73′′.
We consider now the constraints on the Hořava–Lifshitz gravity arising from the Solar System observations of the light deflection by the Sun. The best available data come from long-baseline radio interferometry (Robertson et al. 1991; Lebach et al. 1995), which gives , with ΔLD≤0.0017, where arcsec. The observational constraints of light deflection restrict the value of ω0 to ω0∈(1.1×10−15,1.3×10−15).
(c) Radar echo delay
A third Solar System test of GR is the radar echo delay (Shapiro et al. 1971, 1976). The idea of this test is to measure the time required for radar signals to travel to an inner planet or satellite in two circumstances: (i) when the signal passes very near the Sun and (ii) when the ray does not go near the Sun. The time of travel of light t0 between two planets, situated far away from the Sun, is given by 3.25 where dy is the differential distance in the radial direction, and l1 and l2 are the distances of the planets to the Sun. If the light travels close to the Sun, in the metric given by equation (2.7), the time travel is 3.26 v=e(ν−λ)/2 is the speed of light in the presence of the gravitational field. The time difference is 3.27
Since , where R⊙ is the radius of the Sun, we have 3.28
The first experimental Solar System constraints on time delay have come from the Viking lander on Mars (Shapiro et al. 1971, 1976). In the Viking mission, two transponders landed on Mars and two others continued to orbit round it. The latter two transmitted two distinct bands of frequencies, and thus the Solar coronal effect could be corrected for. However, recently, the measurements of the frequency shift of radio photons to and from the Cassini spacecraft as they passed near the Sun have greatly improved the observational constraints on the radio echo delay. For the time delay of the signals emitted on Earth, and which graze the Sun, one obtains , with ΔRD≤(2.1±2.3)×10−5 (Bertotti et al. 2003).
For the case of the Earth–Mars–Sun system, we have RE=l1=1.525×1013 cm (the distance Earth–Sun) and RP=l2=2.491×1013 cm (the distance Mars–Sun). With these values, the standard general relativistic radar echo delay has the value . With the use of equation (3.28), it follows that the time delay for the KS black hole solution of Hořava–Lifshitz gravity can be represented as 3.29
By introducing a new variable ξ, defined as y=2ξM⊙, and by representing again ω as , we obtain for the time delay the following expression 3.30 where , ξ1=l1/2M⊙ and ξ2=l2/2M⊙. The variation of the time delay as a function of ω0 is represented in figure 4. In the limiting case 4M/ωr3≪1, we obtain again the standard general relativistic result (Shapiro et al. 1971, 1976).
The observational values of the radar echo delay are consistent with the KS black hole solution in Hořava–Lifshitz gravity if ω0∈(2.0199×10−15,2.2000×10−15). The general relativistic value ΔtRD=2.4927×10−4 is obtained for ω0≈4×10−15.
4. Discussion and final remarks
In the present paper, we have considered observational possibilities for testing the KS solution of the vacuum field equations in Hořava–Lifshitz gravity at the level of the Solar System. We have found that this solution can give a satisfactory description of the perihelion precession of the planet Mercury, and of the other gravitational phenomena in the Solar System. The classical tests of GR (perihelion precession, light deflection and radar echo delay) give strong constraints on the numerical value of the parameter ω of the model. The parameter ω, having the physical dimensions of length−2, is constrained by the perihelion precession of the planet Mercury to a value of . The deflection angle of the light rays by the Sun can be fully explained in Hořava–Lifshitz gravity with the parameter ω having the value , while the radar echo delay experiment suggests a value of . Tentatively, and in order to provide a numerical value that could be used in practical calculations, from these values, we can give an estimate of ω as 4.1
The standard deviation in our determination of ω is 3.2×10−26 cm−2, the variance is 9.7×10−26 cm2 and the median of the determined values is 4.5×10−26 cm2. It is interesting to note that the values of ω obtained from the study of the light deflection by the Sun and of the radar echo delay experiment are extremely close, while the perihelion precession of the planet Mercury provides a smaller value. Unfortunately, the observational data on the perihelion precession are strongly affected by the solar oblateness, whose value is poorly known (Mecheri et al. 2004; Pireaux et al. 2007). We have also neglected the solar lense-thirring effect (Iorio et al. in press), as well as the effects of the asteroids. Even so, by taking into account the smallness of the parameter ω, it follows that there is a very good agreement between the numerical values for ω obtained from these three Solar System tests. On the other hand, it is important to note that the light deflection and the radar echo delay observations, which are very similar from a physical point of view, do not give intersecting intervals for ω0, since the value of ω0 obtained from light deflection is in the range ω0∈(1.1×10−15,1.3×10−15), while from the radar echo delay we obtain ω0∈(2.0199×10−15,2.2000×10−15). The values obtained from the light deflection by the Sun are systematically smaller, by a factor of around 2, than the values obtained from radar echo delay observations. This non-intersecting range of values could be explained by the differences in the observational errors for the two effects. While the observational error in light deflection is around 0.0017, the corresponding error in the radar echo delay observations is of the order of 10−5−10−6. The very small error of the radar echo delay allows a very precise determination of the value of ω0. Another possibility for this discrepancy may be related to some intrinsic properties of the model, such as the fact that in Hořava–Lifshitz gravity there is no full diffeomorphism invariance of the Hamiltonian formalism.
In the weak-field and slow-motion approximation, the corrections to the third Kepler Law of a test particle in the KS black hole geometry were obtained in Iorio & Ruggiero (in press a). The corrections were compared with the phenomenologically determined orbital period of the transiting extrasolar planet HD209458b Osiris. The order-of-magnitude lower bound on the parameter ω0 obtained from this study is ω0≥1.4×10−18, as compared with the value ω0≥7×10−16 obtained in the present paper. Tighter constraints are established by the inner planets, for which ω0≥10−15−10−12 (Iorio & Ruggiero in press a). However, in order to obtain a better precision from these data, a full general relativistic study is needed, as well as a significant improvement in the determination of the values of the orbital periods of the exoplanets.
Thus, the gravitational dynamics of the KS solution is determined by the free parameter ω. In order to explain the observational effects in the Solar System, ω must have an extremely small value, of the order of a few 10−26 cm−2. Therefore, the explanation of the classical tests of GR must require a very precise fine-tuning of this constant at the level of the Solar System. It is also very important for future observations to determine whether ω is a local quantity or a universal constant. By assuming that ω is a universal constant, its smallness suggests the possibility that it may have a microscopic origin.
In conclusion, the study of the classical tests of GR provide a very powerful method for constraining the allowed parameter space of the Hořava–Lifshitz gravity solutions, and to provide a deeper insight into the physical nature and properties of the corresponding space–time metrics. Therefore, this opens up the possibility of testing Hořava–Lifshitz gravity by using astronomical and astrophysical observations at the Solar System scale (Iorio & Ruggiero in press a). Of course, this analysis requires general methods to be developed for the high-precision study of the classical tests in arbitrary spherically symmetric space–times. In the present paper, we have provided some basic theoretical tools necessary for the in-depth comparison of the predictions of the Hořava–Lifshitz gravity model with the observational/experimental results.
We would like to thank the two anonymous referees for comments and suggestions that helped us to significantly improve our manuscript. The work of T.H. was supported by the General Research Fund grant no. HKU 701808P of the government of the Hong Kong Special Administrative Region.
- Received September 13, 2010.
- Accepted October 26, 2010.
- This journal is © 2010 The Royal Society