Bending of light in the framework of R + R2 gravity

Bending of light in the framework of R + R² gravity

A. Accioly⁽¹⁾, A.D. Azeredo⁽¹⁾, E.C. de Rey Neto⁽¹⁾ and H. Mukai⁽²⁾

⁽¹⁾Instituto de Física Teórica,

Universidade Estadual Paulista,

Rua Pamplona 145,

01405-900, São Paulo, SP - Brazil

⁽²⁾Departamento de Física,

Fundação Universidade Estadual de Maringá,

Av. Colombo 5790, 87020-900 Maringá, Pr, Brazil

e-mail: [e-mail]

Received 15 September, 1998

We present a tree level approach to the issue of the deflection of photons by the gravitational field of the Sun, treated as an external field, on the basis of R + R² gravity. We show that the deflection angle of a photon grazing the surface of the Sun is exactly the same as that given by general relativity. An explanation for this strange coincidence is provided.

I. Introduction

One of the simplest and most enduring ideas for extending general relativity is to include in the Einstein's gravitational action terms involving higher powers of the curvature tensor. These extra terms neither break the invariance of the action under general coordinate transformations nor can be excluded by any known principle. If, for instance, we include only a term proportional to R², the resulting action is[1]

where k² = 32 pG, and a is a dimensionless parameter. Since the quantity R involves two derivatives acting on the metric and in an interaction each derivative becomes a factor of the momentum transfer k, or of the inverse distance scale , we may say that R is of order , while R² is of order , respectively. Thus, at small enough energies, or at very long distances, the quadratic term is negligible and R + R² gravity reduces to general relativity. Accordingly, for small values of the energy E of incident radiation ( E ~ 0 ) we expect the measurements of the solar gravitational deflection to be in excellent agreement with general relativity. As a matter of fact, in the measurements of the solar gravitational deflection of the radio waves ( E ~ 1cm^{- 1} ), Fomalont and Sramek [2] found a value of 1.76 ±0.016 arcsec for the deflection at the solar limbs (the prediction of general relativity is 1.75 arcsec). Therefore, a non negligible contribution of the quadratic term to the solar gravitational deflection of starlight ( E ~ 10⁵cm^{- 1} ) is to be expected. Incidentally, the mean value of the deflection of light at the solar limb obtained by averaging over the measurements made on all eclipses expeditions to date is 2.04 arcsec. These considerations lead us to conjecture that R + R² gravity should provide a value for the deflection angle of starlight which passes the Sun's limb on its way to the Earth nearer to the experimental value than that given by general relativity. We address this question here. To do that we compute the scattering of a photon by the Sun's gravitational field, treated as an external field, on the basis of R + R² gravity.

We show in Sec. II that only in the absence of tachyons in the dynamical field does R + R² gravity give an acceptable newtonian limit. In Sec. III we compute the gravitational field of the Sun, treated as a point particle, in the weak field approximation. In Sec. IV we calculate the cross-section for the scattering of a photon by an external gravitational field and show from this result that the deflection angle of a photon grazing the surface of the Sun predicted by R + R² gravity is exactly the same as that provided by general relativity. An explanation for this strange coincidence is given in Sec. V.

Natural units are used throughout (c = = 1). In our notation the signature is (+ - - -). The curvature tensor is defined by R^a_bgd = -¶_dG^a_bg+¼, the Ricci tensor by R_mn = R^a_mna, and the curvature scalar by R = g^mnR_mn, where g_mn is the metric tensor.

II. Absence of tachyons as a condition for R + R² gravity to give an acceptable newtonian limit

The effective nonrelativistic potential for the gravitational interaction of two identical massive bosons of zero spin (see Fig. 1) is

Figure 1.
Feynman diagram for the gravitational interaction of two identical massive bosons of zero spin.

whereupon

where M_NR is the nonrelativistic limit of the Feynman amplitude for the process depicted in Fig. 1. On the other hand, the Lagrangian density for the interaction of gravity with a free, massive scalar field f, is

The quantum fluctuations of the gravitational field may be expanded about a smooth background metric, which in our case is flat spacetime

From (3) and (4), we obtain

where the Lagrangian density for the free scalar field has been omitted. From the previous expression the Feynman rule for the elementary vertex may readily be deduced. It is shown in Fig. 2. In momentum space, the free propagator for R+R² gravity, in the de Donder (harmonic) gauge, is (see ^{appendix A})

Figure 2.
The relevant Feynman rule for boson-boson interaction.

The Feynman amplitude for the reaction displayed in Fig. 1 is then given by

where

which in the nonrelativistic limit reduces to

Eq. (5) can be rewritten as

where

From (6) and (2) we promptly obtain

where

From a mathematical point of view integral (7) makes sense only if a is positive , which corresponds to the absence of tachyons in the dynamical field. Assuming that a > 0, integral (7) can be easily evaluated by the method of contour integration. The result is

Therefore

The associated nonrelativistic gravitational potential for R + R² gravity is then given by

Note that (8) gives an acceptable newtonian limit since there is only a falling exponential and not a oscillating term at infinity.

III. The gravitational field outside the Sun in the framework of R + R² gravity and in the weak field approximation

The action for R + R² gravity theory is given by

where L_M is the Lagrangian density for the usual matter. The corresponding field equation is[3]

where

In the weak field approximation this equation reduces to

where

In obtaining the previous equation we have used the harmonic condition g_mn ^{, n} = 0.

Contracting (10), we get

Let us assume that the Sun is a point particle of mass M located at r = 0. The corresponding energy-momentum tensor is given by

From (11) and (12), we promptly obtain

where M₀² is supposed to be positive since we are assuming the absence of tachyons in the dynamical field (see the previous section). This equation is solved in detail in Ref. [4]. The solution is

From (9) and (12), we have

whose solution is

Note that (see the preceding section).

From (9) and (12), we finally get

whose solution can be written as

where

Of course, h_i 0 = 0. Therefore, the gravitational field outside the Sun takes the form

Since in the next section we will need the momentum space gravitational field, i. e. ,

we use the remainder of this section to evaluate h_mn ( k ). This calculation is rather involved. Using much algebra and the integrals listed in ^{appendix B}, some of which only exist as distributions, we arrive at the following result

where , and k º |k |.

IV. Prediction of R + R² gravity for the deflection angle of a photon grazing the Sun's surface

Let us now consider the scattering of a photon by the gravitational field of the Sun, treated as an external field.

The Lagrangian function of the system is

The corresponding photon-external-gravitational-field vertex is shown in Fig. 3. Therefore, the Feynman amplitude for the process displayed in Fig. 3 is of the form

Figure 3.
The photon-external-gravitational-field vertex.

whereupon

where ( p ) and ( p¢) are the polarization vectors for the initial and final photons, respectively.

The unpolarized cross-section for the process is proportional to

where we have averaged over initial polarizations and summed over final polarizations. Hence

Since , the momentum space gravitational field can be rewritten as

Therefore,

where E is the energy of the incident photon and q is the scattering angle. The unpolarized cross-section for the process in hand can be obtained from the expression [5]

Thus, the unpolarized cross-section for the scattering of a photon by the Sun's gravitational field, treated as an external field, on the basis of R + R² gravity, is given by

which, for small angles, reduces to

On the other hand,

For small q, we get

From (14) and (15) we then have

So, we come to the conclusion that the prediction of R + R² gravity for the deflection angle of a photon passing near to the Sun is exactly the same as that given by Einstein's theory, namely,

Actually, it is a paradox that two quite different gravitational theories could lead to the same value for the solar gravitational deflection. Where such a strange coincidence came from? We discuss this important matter in the next section.

V. A general result

We have argued that the quadratic term of R+R² gravity theory should give a non negligible contribution to the gravitational solar deflection of starlight (E ~ 10⁵cm^-1). However, a careful calculation at the tree level showed that the deflection angle of a photon passing near to the Sun's surface is the same as that predicted by Einstein's theory. In other words, the quadratic term does not contribute anything to the gravitational solar deflection. How can we explain this unexpected result? First of all, we note that at the tree level, since

whereupon

where (r) is the solution of the linearized Einstein's equation supplemented by the usual harmonic coordinate condition, we may write the Feynman amplitude as

where

Of course, . Therefore,

which clearly shows why the two theories lead to the same result for the solar gravitational deflection.

At the classical level, in turn, the field equation for R+R² gravity, in the weak field approximation and in the gauge , where , is given by

In this equation R must be replaced by its respective first-order expression:

In this gauge, the Sun's gravitational takes the form

where

Neglecting the product of by f, the metric can be written as

Thus, in the absence of tachyons ( a > 0 ) and assuming that 1 +kF > 0 (this condition is necessary for the conformal transformation not to invert the signature of the metric) , the linearized R + R² gravity theory is conformally related to Einstein's linearized theory.

The previous result clearly shows why we arrive at the same value for the solar gravitational deflection whether we use R + R² gravity or Einstein's gravity.

To conclude, we call attention to the fact that Whitt[6] and Teyssandier[7] studied the equivalence of the above theories in a quite different context.

Acknowledgment. E. C. de Rey Neto is very grateful to Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for financial support.

Appendix A: The propagator for R + R² gravity

In order to calculate the propagator for R+R² gravity, we must first write out the Lagrangian density in a purely quadratic form in the gravitational field h^mn. Going over to momentum space and using a substantial amount of Lorentz algebra on symmetric rank-two tensors, we arrive at the following expression for L = L_g+L_gf, with the gauge fixing Lagrangian density given by ,

whereupon

where

Since P^{mn, rq} is a fourth-rank tensor dependent only on the 4-vector k^a, Lorentz-covariance therefore imposes the following general form for its inverse

Now, taking into account that

we promptly obtain

The graviton propagator is then given by

Appendix B: A table of integrals for the calculation of h_mn ( k ) (Some of the integrals below only exist as distributions)

References

[1]
• J. H. Kung, Phys. Rev. D 52, 6922 (1995).
[2]
• E. B. Fomalont and R. A. Sramek, Astrophys. J. 199, 749 (1975); Phys. Rev. Lett. 36, 1475 (1976); Comm. Astrophys. 7, 19 (1977).
[3]
• A. J. Accioly, Rev. Bras. Fis. 18, 593 (1988).
[4]
• Antonio Accioly and Hatsumi Mukai, Braz. J. Phys. 28, 35 (1998).
[5]
• C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions (Addison-Wesley Publishing Company, 1996); M. Kaku, Quantum Field Theory - A Modern Introduction (Oxford University Press, 1993); M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley Publishing Companhy, 1996).
[6]
• B. Whitt, Phys Lett. B 145, 176 (1984).
[7]
• P. Teyssandier, Class. Quantum Grav. 6, 219 (1989).

• [2] E. B. Fomalont and R. A. Sramek, Astrophys. J. 199, 749 (1975); Phys. Rev. Lett. 36, 1475 (1976);
• [5] C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions (Addison-Wesley Publishing Company, 1996); M. Kaku, Quantum Field Theory - A Modern Introduction (Oxford University Press, 1993);

appendix A

appendix B

Publication Dates

• Publication in this collection
  26 Aug 1999
• Date of issue
  Dec 1998

History

• Received
  15 Sept 1998