there were also broader objections to the equivalence betweenf(R) and scalar-tensor gravity [75].
In this chapter, we will show that even though the Schwarzschild-de Sitter metric is a vacuum solution in f(R) gravity, it does not correspond to the solution around a spherically symmetric massive body. The solution for the Solar System is identical to the spacetime derived using the corresponding scalar-tensor theory. We will work directly with the field equations in the metric formalism, where the field equations are obtained by varying the action with respect to the metric and treating the Ricci scalar as a function of the metric. The Palatini formalism, which treats the Ricci scalar as a function of the connection and varies the action with respect to the connection and the metric independently, yields different field equations forf(R) gravity and has been studied extensively elsewhere (e.g. Refs. [118, 119, 120, 121, 122]).
We begin in Section 3.2 by considering 1/Rgravity in detail. We then generalize this analysis to a broad class off(R) gravities, namely those theories that admit a Taylor expansion off(R) around the background value of the Ricci scalar. In Section 3.3, we solve the linearized field equations around a spherical mass and find that the solution in the Solar System is in agreement with the solution obtained using the equivalent scalar-tensor theory. Whenf(R) satisfies a condition that is analogous to the scalar field being light in the equivalent scalar-tensor theory, and nonlinear effects are negligible, the resulting spacetime is incompatible with Solar System tests of general relativity. In Section 3.4, we consider how our analysis applies to severalf(R) gravity theories, including general relativity. This particular example illustrates the connection between f(R) gravity and general relativity and clarifies the requirements for a general relativistic limit of an f(R) theory. Section 3.5 is devoted to a particularly interesting f(R) theory, proposed by Ref. [79], that is compatible with Solar System gravitational tests. We show how this theory uses a nonlinear effect to mask its difference from general relativity. Finally, we summarize our conclusions in Section 3.6 and list a set of conditions that, when satisfied by a givenf(R) theory, imply that the theory is ruled out by Solar System tests of general relativity.
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We begin by using the trace of the field equation to determine the Ricci scalar R. Contracting Eq. (3.2) with the inverse metric yields
µ4 R2−R
3 +µ4
R = 8πGT
3 , (3.3)
whereT ≡gµνTµν.
The constant-curvature vacuum solution is obtained by setting T = 0 and∇µR= 0. It isR2= 3µ4, corresponding to the de Sitter spacetime with Hubble parameterH2 =µ2/(4√
3), equivalent to the general-relativistic vacuum solution with a cosmological constant Λ = 3H2 =√
3µ2/4. The metric for this spacetime can be written as a static spherically-symmetric spacetime:
ds2=− 1−H2r2
dt2+ 1−H2r2−1
dr2+r2dΩ2. (3.4) To match the observed acceleration of the universe, the effective cosmological constant must be set to Λ∼µ2∼H2∼10−56cm−2.
We now consider the spacetime in the Solar System in this theory. First of all, the distances (∼1013 cm) in the Solar System are tiny compared with the distanceµ−1 ∼1028 cm, so µr≪1 everywhere in the Solar System. Moreover, the densities and velocities in the Solar System are sufficiently small that we can treat the spacetime as a small perturbation to the de Sitter spacetime.
The spacetime should also be spherically symmetric and static. The most general static spherically- symmetric perturbation to the vacuum de Sitter spacetime given by Eq. (3.4) can be written
ds2=−
1 +a(r)−H2r2 dt2+
1 +b(r)−H2r2−1
dr2+r2dΩ2, (3.5) where the metric-perturbation variablesa(r), b(r)≪1. In the following, we work to linear order in aandb, and also recall thatµr≪1. However,aandbarenotnecessarily small compared withµr.
We now return to the trace of the field equation, given by Eq. (3.3), and solve it for the Ricci scalarR(r) in the presence of the Sun. We write the trace equation in terms of a new function,
c(r)≡ −1 3 + µ4
R2(r), (3.6)
and demand that c(r)→0 asr→ ∞ so thatRapproaches its background value of √
3µ2 far from the source of the perturbation. Therefore,c(r) parameterizes the departure ofR from the vacuum solution, and we anticipate that c(r) will be the same order in the perturbation amplitude as the metric perturbationsa(r) andb(r). In terms ofc(r), Eq. (3.3) becomes an exactequation,
c(r) + µ2c q
c+13
= 8πG
3 T. (3.7)
In the Newtonian limit appropriate for the Solar System, the pressurepis negligible compared to the energy densityρ, and soT =−ρ. Neglecting terms that are higher order ina(r),b(r), andµ2r2, we are able to rewrite Eq. (3.7) as
∇2c+√
3µ2c=−8πG
3 ρ, (3.8)
where ∇2 is the flat-space Laplacian operator. Note that in writing this equation, which is linear in c(r), we have also neglected higher-order terms in c(r). Below, we will check that the solutions we obtain have c(r) ≪ 1 everywhere, consistent with our assumptions. The Green’s function for Eq. (3.8) is−cos(31/4µr)/(4πr). Convolving this with the density gives us the solution to Eq. (3.8).
However, we are restricting our attention to the region whereµr≪1, so the Green’s function reduces to that for the Laplacian operator. Therefore the equation we need to solve is ∇2c=−(8πGρ)/3.
Integrating the right-hand side over a spherical volume of radius r gives us −8πGm(r)/3, where m(r) is the mass enclosed by a radiusr. Using Gauss’s law to integrate the left-hand side gives us 4πr2c′(r), where the prime denotes differentiation with respect to r. Thus, the equation for c(r) becomes
dc
dr =−2Gm(r)
3r2 [1 +O(µr)]. (3.9)
Integrating Eq. (3.9) and using the boundary condition thatc→0 asr→ ∞gives us the solution
c(r) = 2 3
GM r
[1 +O(µr)], forr > R⊙. (3.10) We also note that integration of the equation forc′(r) to radiir < R⊙ inside the star implies that the scalar curvatureRremains of orderµ2, even inside the star. We thus see thatc≪1, so we were justified in using the linearized equation forc(r).
This solution forc(r) implies that R=√
3µ2
1−GM r
, forr > R⊙. (3.11) We have thus shown thatR is not constant outside the star and have already arrived at a result that is at odds with the constant-curvature Schwarzschild-de Sitter solution. Notice that had we (incorrectly) usedρ= 0 in Eq. (3.8), then the equations would have admitted the solutionc(r) = 0;
i.e., the constant-curvature solution. However, this would be incorrect, because even thoughρ= 0 at r > R⊙, the solution to the differential equation at r > R⊙ depends on the mass distribution ρ(r) atr < R⊙. In other words, although the Schwarzschild-de Sitter solution is a static spherically- symmetric solution to the vacuum Einstein equations, it is not the solution that correctly matches onto the solution inside the star.
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The solution forR both inside and outside the star is (to linear order inc), R=√
3µ2
1−3 2c(r)
. (3.12)
Clearly, 1/Rgravity produces a spacetime inside the star that isverydifferent from general relativity.
This result shows that in this theory one should not assume thatR≃8πGρ; this has lead to some confusion [123, 124, 125].
To proceed to the solutions for a(r) and b(r), we rearrange the field equation for 1/R gravity [Eq. (3.2)] to obtain equations,
Rµν =
1 + µ4 R2
−1
8πGTµν+1 2
1− µ4
R2
Rgµν−µ4(gµν∇α∇α− ∇µ∇ν)R−2
, (3.13) for the Ricci tensor in terms of the Ricci scalar. When the expression forRobtained from the trace equation is inserted into the right-hand side, we obtain equations for the nonzero components of the Ricci tensor,
Rtt = 3H2−6πGρ−3
4∇2c, (3.14)
Rrr = 3H2−3c′(r)
2r , (3.15)
Rθθ=Rφφ = 3H2−3 4
c′(r)
r +c′′(r)
, (3.16)
where we have neglected terms of orderµ2c,Gρcandc2in all three expressions.
For the perturbed metric given by Eq. (3.5), thettcomponent of the Ricci tensor is (to linear order in small quantities)Rtt= 3H2−(1/2)∇2a(r). Applying∇2c=−(8πGρ)/3 to Eq. (3.14) leaves us with an equation fora(r),
1
2∇2a= 4πGρ, (3.17)
plus terms that are higher order in GM/r andµr. The solution to this equation parallels that for c(r); it is
da
dr = 2Gm(r)
r2 (3.18)
both inside and outside the star. Outside the star, this expression may be integrated, subject to the boundary conditiona(r)→0 asr→ ∞, to obtain the metric perturbation,
a(r) =−2GM
r , r > R⊙, (3.19)
exterior to the star. Note that this recovers the Newtonian limit for the motion of nonrelativistic bodies in the Solar System, as it should.
Therrcomponent of the Ricci tensor is (to linear order in small quantities)Rrr= 3H2−(b′/r)− (a′′/2). Given our solution for a′(r) and c′(r) = −(2/3)Gm(r)/r2, Eq. (3.15) becomes a simple differential equation forb(r),
db
dr = Gm(r)
r2 −Gm′(r) r
= d
dr
−Gm(r) r
. (3.20)
Integrating this equation subject to the boundary conditionb(r)→0 asr→ ∞gives an expression forb(r) that is applicable both inside and outside the star:
b(r) =−Gm(r)
r . (3.21)
This expression for b(r) and Eq. (3.18) fora′(r) also satisfy Eq. (3.16) for the angular components of the Ricci tensor. The Ricci scalar [Eq. (3.11)] is recovered from the Ricci tensor components if terms higher order inO(µr2GM/r) are included in our expressions fora(r) andb(r).
The linearized metric outside the star thus becomes ds2=−
1−2GM
r −H2r2
dt2+
1 + GM
r +H2r2
dr2+r2dΩ2. (3.22)
To linear order inGM/r andH2r2, this metric is equivalent to the isotropic metric ds2=−
1−2GM
r −H2r2
dt2+
1 + GM r −1
2H2r2
[dr2+r2dΩ2]. (3.23) The PPN parameterγis defined by the metric,
ds2=−
1−2GM r
dt2+
1 +2γGM r
[dr2+r2dΩ2]. (3.24) Given that Hr≪1 in the Solar System, we find that γ = 1/2 for 1/Rgravity, in agreement with Chiba’s claims [72, 126], and prior calculations in scalar-tensor gravity theories: e.g., Refs. [115, 127].
We note that recent measurements giveγ= 1 + (2.1±2.3)×10−5 [116, 117].
It has been noted that Birkhoff’s theorem [128], which states that the unique static spherically- symmetric vacuum spacetime in general relativity is the Schwarzschild spacetime, is lost in 1/R gravity, and that there may be several spherically-symmetric vacuum spacetimes. This is true, and the absence of Birkhoff’s theorem implies it is not sufficient to find any static, spherically-symmetric solution to the vacuum field equations 1/R gravity when attempting to describe the spacetime in the Solar System. What we have shown here is that the Solar System spacetime is determined uniquely by matching the exterior vacuum solution to the solution inside the Sun. When this is
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done correctly, it is found that the theory predicts a PPN parameterγ = 1/2, in gross violation of the measurements [116, 117], which requireγ to be extremely close to unity.