String corrections to the Sagnac effect

Abstract

In an effort to investigate string effects in physical observations, we have analyzed the rotating Kerr–Sen metric in a Sagnac type experiment and have deduced exact expressions for the delay. For an Earth bound configuration, it turns out that a correction to the basic Sagnac delay by an order of ∼10⁻¹⁴ s leads to a terrestrial dilatonic charge of amount ∼10²⁴ esu, a value nearly 200 times larger than the electronic charge of the Earth's magnetosphere.

Introduction

Low energy effective field theory describing heterotic string theory has now become an indispensable part of the frontiers of theoretical physics [1]. An interesting result is that black hole solutions exist also in the string theory and that they exhibit qualitatively different properties than those of Einstein's general relativity [2]. A rotating black hole solution, that reduces to the Kerr solution for a constant dilaton field, has been constructed and analyzed by Horne and Horowitz [3].

A more general classical exact solution has been found by Sen [4] which we refer to here as the Kerr–Sen metric. The action underlying the theory has a U(1) gauge symmetry and contains antisymmetric tensor gauge field. Also, 6 of the 10 dimensions are compactified to a suitable manifold, but the resulting massless fields are not included in the action. The absence of these fields enhances the possibility of the black hole nature of the solution; otherwise, naked singularities could arise. Kerr–Sen solution describes a rotating black hole carrying finite amount of charge and angular momentum and it differs from the Horowitz–Horne black hole even in the limit of small angular momentum. This difference arises due to the coupling of the antisymmetric tensor gauge field to the Chern–Simons term in the action considered by Sen [4]. These, and other developments taken together, indicate that there have been tremendous theoretical advances in the understanding and utility of the string theory. However, relatively much less is discussed in the literature as to how a black hole in the string theory could possibly affect physical observations in practice. It has been shown by Gegenberg [5] that static spherically symmetric solutions do not lead to string effects in the PPN approximation of the solar system scenario but that, he conjectured, rotating solutions might lead to observable string effects, however tiny.

The present Letter aims to undertake an investigation precisely in this direction. For this purpose, we consider the Kerr–Sen metric and examine how the black hole parameters appear in the correction terms in a Sagnac-type experiment. We shall find exact expressions for the time delay by following the procedure of Tartaglia [6] which we had also adopted in our recent investigation of the Brans–Dicke correction factors for different types of orbits [7]. An estimate of the possible terrestrial dilatonic charge and some remarks are also added.

We have chosen the Sagnac effect because of its simplicity and its easy adaptability to rotating sources. The effect stems from the basic physical fact that the round trip time of light around a closed contour, when the source is fixed on a turntable, depends on the angular velocity, say $\text{Ω}$, of the turntable. Using special theory of relativity, and assuming $\text{Ωr⪡c}$, one obtains the proper time difference δτ_s when the two beams meet again at the starting point as [6]

$$\text{δτ}{}_{\mathrm{s}}\text{≅}\text{4Ω}\text{c}{}^2\text{S,}$$

where c is the vacuum speed of light, S (=πr²) is the projected area of the contour perpendicular to the axis of rotation. It is a real physical effect in the sense that it does not involve any arbitrary synchronization convention that is required between two distant clocks [8]. The effect is also universal as it manifests not only for light rays but also for all kinds of waves including matter waves [9]. Formula (1) has been tested to a good accuracy and the remarkable degree of precision attained lately by the advent of ring laser interferometry raises the hope that measurements of higher-order corrections to this effect might be possible in near future [6], [10].

The string theory effective action in 4 dimensions, considered by Sen [4], is

$$\text{S=−∫d}{}^4\text{x}\text{−g}\text{e}{}^{\mathrm{-\Phi }}\text{−R+}\text{1}\text{12}\text{H}{}_{\mathrm{\mu \nu \rho }}\text{H}{}^{\mathrm{\mu \nu \rho }}\text{−g}{}^{\mathrm{\mu \nu }}\text{∂}{}_{\mathrm{\mu }}\text{Φ∂}{}_{\mathrm{\nu }}\text{Φ+}\text{1}\text{8}\text{F}{}_{\mathrm{\mu \nu }}\text{F}{}^{\mathrm{\mu \nu }}\text{,}$$

where g_μν is the metric that arises naturally in the σ-model, R is the Ricci scalar, F_μν≡∂_μA_ν−∂_νA_μ is the field strength of the Maxwell field A_μ, Φ is the dilaton field, and

$$\text{H}{}_{\mathrm{\mu \nu \rho }}\text{≡∂}{}_{\mathrm{\mu }}\text{B}{}_{\mathrm{\nu \rho }}\text{+∂}{}_{\mathrm{\nu }}\text{B}{}_{\mathrm{\rho \mu }}\text{+∂}{}_{\mathrm{\rho }}\text{B}{}_{\mathrm{\mu \nu }}\text{−}\text{Ω}{}_3\text{(A)}{}_{\mathrm{\mu \nu \rho }}\text{,}$$

where B_μν is the antisymmetric tensor gauge field and

$$\text{Ω}{}_3\text{(A)}{}_{\mathrm{\mu \nu \rho }}\text{≡}\text{1}\text{4}\text{(A}{}_{\mathrm{\mu }}\text{F}{}_{\mathrm{\nu \rho }}\text{+A}{}_{\mathrm{\nu }}\text{F}{}_{\mathrm{\rho \mu }}\text{+A}{}_{\mathrm{\rho }}\text{F}{}_{\mathrm{\mu \nu }}\text{)}$$

is the gauge Chern–Simons term. The Einstein frame metric is obtained from the relation

$$\text{g}{}_{\mathrm{\mu \nu }}{}^{\mathrm{(E)}}\text{=e}{}^{\mathrm{-\Phi }}\text{g}{}_{\mathrm{\mu \nu }}\text{.}$$

For our purposes, we recast the Einstein frame Kerr–Sen metric into a form that closely resembles the familiar Kerr solution in Boyer–Lindquist coordinates. The result is (G=c=1)

$$\text{dτ}{}^2\text{=}\text{1−}\text{2Mρ}\text{Σ}\text{dt}{}^2\text{−Σ}\text{dρ}{}^2\text{Δ}\text{+dθ}{}^2\text{−}\text{ρ(ρ+ξ)+a}{}^2\text{+}\text{2Mρa}{}^2\text{sin}{}^2\text{θ}\text{Σ}\text{sin}{}^2\text{θ}\text{dϕ}{}^2\text{+}\text{4Mρa}\text{sin}{}^2\text{θ}\text{Σ}\text{dt}\text{dϕ,}$$

$$\text{Φ=−}\text{ln}\text{Σ}\text{ρ}{}^2\text{+a}{}^2\text{cos}{}^2\text{θ}\text{,}\text{A}{}_{\mathrm{\varphi }}\text{=−}\text{2}\text{2}\text{aρQ}\text{sin}{}^2\text{θ}\text{Σ}\text{,}\text{A}{}_{\mathrm{t}}\text{=}\text{2}\text{2}\text{ρQ}\text{Σ}\text{,}\text{B}{}_{\mathrm{t\varphi }}\text{=}\text{aρQ}{}^2\text{sin}{}^2\text{θ}\text{MΣ}\text{,}\text{Σ=ρ(ρ+ξ)+a}{}^2\text{cos}{}^2\text{θ,}\text{Δ=ρ(ρ+ξ)+a}{}^2\text{−2Mρ,}\text{ξ=}\text{Q}{}^2\text{M}\text{.}$$

The metric describes a black hole with mass M, dilatonic charge Q, angular momentum aM, and magnetic dipole moment aQ. For Q=0, the metric reduces to the Kerr solution of GR and for a=0, it reduces to the Gibbons–Garfinkle–Horowitz–Strominger black hole solution [2] with the redefinition: ρ→r−ξ. Using metric (5), we proceed to calculate the proper Sagnac delay for three types of source/receiver orbits: equatorial, polar and geodesic circular orbits.