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Gravitational Time Delay of Light for Various Models of Modified Gravity Physics Letters B 661 (2008) 78–81 www.elsevier.com/locate/physletb Gravitational time delay of light for various models of modified gravity Hideki Asada Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan Received 14 December 2007; received in revised form 30 January 2008; accepted 1 February 2008 Available online 7 February 2008 Editor: M. Cveticˇ Abstract We reexamined the gravitational time delay of light, allowing for various models of modified gravity. We clarify the dependence of the time delay (and induced frequency shift) on modified gravity models and investigate how to distinguish those models, when light propagates in static spherically symmetric spacetimes. Thus experiments by radio signal from spacecrafts at very different distances from Sun and future space-borne laser interferometric detectors could be a probe of modified gravity in the solar system. © 2008 Elsevier B.V. All rights reserved. PACS: 04.80.Cc; 04.50.+h; 95.30.Sf; 95.36.+x The nature of dark energy and dark matter has become a cen- effect has successfully tested the Einstein’s theory [7]. A signif- tral issue in modern cosmology. Recent observations such as the icant improvement was reported in 2003 from Doppler track- magnitude-redshift relation of type Ia supernovae (SNIa) [1] ing of the Cassini spacecraft on its way to the Saturn, with and the cosmic microwave background (CMB) anisotropy by γ − 1 = (2.1 ± 2.3) × 10−5 [8]. Here, γ is one of parame- WMAP [2] strongly suggest a certain modification, in what- ters in the parameterized post-Newtonian (PPN) formulation of ever form, in the standard cosmological model. We are forced gravity [5]. The bending and delay of photons by the curvature to add a new component into the energy–momentum tensor in of spacetime produced by any mass are proportional to γ + 1, the Einstein equation or modify the theory of general relativity where γ is unity in general relativity but zero in the Newtonian itself [3]. Indeed, there have been a lot of proposals motivated theory, and the quantity γ −1 is thus considered as a measure of by, for instance, scalar tensor theories, string theories, higher di- a deviation from general relativity. The sensitivity in the Cassini mensional scenarios and quantum gravity. (For recent reviews experiment approaches the level at which, theoretically, devi- of modified gravity models inspired by the dark energy obser- ations 10−6–10−7 are expected in some cosmological models vation, e.g., [4].) Therefore, it is of great importance to obser- [9,10]. Therefore, it is important to investigate the Shapiro time vationally test these models. delay with such a high accuracy. The theory of general relativity has passed “classical” tests, In addition to the above theoretical motivation, there are ad- such as the deflection of light, the perihelion shift of Mercury vances in technologies concerning the high precision measure- and the Shapiro time delay, and also a systematic test using the ment of time and frequency such as optical lattice clocks [11] remarkable binary pulsar “PSR 1913 + 16” and several binary and attoseconds (10−18 s) laser technologies [12].ASTROD pulsars now known [5]. In the twentieth century, these tests project with three spacecrafts aims at measuring γ at the level proved that the Einstein’s theory is correct with a similar ac- of 10−9 [13]. curacy of 0.1%. The purpose of this Letter is to clarify the dependence of Since the time delay effect along a light path in the grav- the time delay (and induced frequency shift) on modified grav- itational field was first noticed in 1964 by Shapiro [6],this ity models and investigate how to distinguish those models by using the Shapiro time delay. An important point in this Letter is that we allow for various modified gravity theories beyond E-mail address: [email protected]. the scope of the PPN formulation. Introducing a new energy or 0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2008.02.006 H. Asada / Physics Letters B 661 (2008) 78–81 79 length scale (e.g. extra dimension scale) may make changes in The Cassini experiment has put the tightest constraint on the functional forms of the gravitational field. Thus it is worthwhile solar gravity, especially near the solar surface with the accuracy − to investigate how to probe such a modified functional form, by of 10 5 [8]. This implies that deviations in A(r) and B(r) −5 −10 m using the light propagation in the solar system. Throughout this must be less than 10 × 2M/r ∼ 10 , that is, |Amr |, n −10 Letter, we take the units of G = c = 1. |Bnr| < 10 . In this Letter, we assume that the electromagnetic fields We consider the round-trip time between pulse transmission propagate in four-dimensional spacetimes (even if the whole and echo reception, denoted by T . The pulse is emitted from spacetime is higher dimensional). Thus photon paths follow Earth at rE , and reflected at rR. null geodesics (as the geometrical optics approximation of Up to the linear order in M, An and Bn, T is expressed as Maxwell equation). = 2 − 2 + 2 − 2 We shall consider a static spherically symmetric spacetime, T 2 rE r0 rR r0 in which light propagates, expressed as + 2 − 2 + 2 − 2 2 2 2 2 2 rE rE r0 rR rR r0 ds =−A(r) dt + B(r)dr + r dΩ , (1) + 2M 2ln + 2ln r0 r0 where r and dΩ2 denote the circumference radius and the met- r − r r − r ric of the unit 2-sphere, respectively. The functions A(r) and + E 0 + R 0 + δt. + + (5) B(r) depend on gravity theories. rE r0 rR r0 The time lapse along a photon path is obtained as The extra time delay induced by a correction to general relativ- r ity is expressed as dr B(r) 1 t(r,r ) = , (2) R R 0 b A(r) E R A(r0) − A(r) n+1 r0 2 2 δt = r + dR r0 r 0 1 1 where b and r denote the impact parameter and the closest 0 n+3 − n+1 + n+1 point, respectively. Their relation is b2 = r2/A(r ). R 2R R R 0 0 × −An + Bn √ , (6) According to a concordance between solar-system experi- (R2 − 1)3/2 R2 − 1 ments and the theory of general relativity, we can assume that where we define non-dimensional radial coordinates as R ≡ the spacetime is expressed as the Schwarzschild metric (rig- r/r0, RE ≡ rE/r0 and RR ≡ rR/r0. For a radar tracking of orously speaking, its weak field approximation) with a small a spacecraft such as Cassini, rE and rR are of the order of perturbation induced by modified gravity. For practical calcu- 8 1AU(∼ 10 km), and r0 is several times of the solar radius lations, we keep only the leading term at a few AU in the (R ∼ 105 km). Eq. (6) can be rewritten by using special func- corrections. Namely, A(r) and B(r) are approximated as tions, though it seems less informative. Therefore, we take ex- 2M pansions of Eq. (6) in r0 because of rE,rR r0.Forn = 1, we A(r) ≈ 1 − + A rm, (3) r m obtain − 2M n Bn An n+1 n+1 B(r) ≈ 1 + + Bnr , (4) δt = r + r r n + 1 E R where M denotes the mass of the central body. Here, A , B , B + A − − − m n + n n rn 1 + rn 1 − 2rn 1 r2 + O r4 , (7) m and n rely on a theory which we wish to test. For simplicity, 2(n − 1) E R 0 0 0 we assume m = n>0, which corresponds to a wide class of whereas the second term of R.H.S. becomes (B +A ) ln(r r / theories of gravity. n n E R r2)r2/2forn = 1. Examples of modified gravity theories are as follows. 0 0 It is convenient to use the relative change in the frequency, (1) n = 1/2, A =−2B =±2 M/r2 for DGP model with n n c which is caused by the gravitational time delay [18], because r is the extra scale within which gravity becomes five- c √ the Doppler shift due to the receiver’s motion has no effect ow- dimensional [14].(2)n = 3/2, A = (2/3)m2 2M/13 and √ n g ing to the cancellation at both the receipt and emission of radio =− 2 Bn mg 2M/13 with graviton mass mg for one of mas- signal [18]. This frequency shift is defined as y =−d(T)/dt. = =− =− sive gravity models [15,16].(3)n 2, An Bn Λ/3for Indeed, the frequency shift was used by the Cassini experiment. the Schwarzschild–de Sitter spacetime, that is, general relativ- For a case of b rE,rR, which is valid for the Cassini experi- ity with the cosmological constant Λ as a possible candidate for ment, the general relativistic contribution is expressed as [5] the dark energy, though this is not a manifest modification of M db gravity. The solar system experiments are not sensitive to this y = 4 . (8) model with Λ ∼ 10−52 m−2 [17]. Here, it should be noted that GR b dt the examples (1) and (2) give conformally flat spacetimes (in We pay attention to the extra contribution due to modified the weak field approximation) and their conformal factors gen- gravity. For n = 1, the extra frequency shift becomes erate the gravitational time delay (and induced frequency shift), A + B − − − db though the null geodesic in any conformally flat spacetime is δy =− n n rn 1 + rn 1 − (n + 1)rn 1 b , (9) mapped into that in the Minkowski one.
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