String corrections to the Sagnac effect
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 , of the turntable. Using special theory of relativity, and assuming , one obtains the proper time difference δτs when the two beams meet again at the starting point as [6] 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 where gμν is the metric that arises naturally in the σ-model, R is the Ricci scalar, Fμν≡∂μAν−∂νAμ is the field strength of the Maxwell field Aμ, Φ is the dilaton field, and where Bμν is the antisymmetric tensor gauge field and is the gauge Chern–Simons term. The Einstein frame metric is obtained from the relation 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) 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.
Section snippets
Equatorial orbit
Suppose that the source/receiver of two oppositely directed light beams is moving around an uncollapsed normal gravitating body, along a circumference at a radius ρ=R=const, on the equatorial plane θ=π/2. Suitably placed mirrors send back to their origin both beams after a circular trip about the central rotating body. Let us further assume that the source/receiver is moving with uniform orbital angular speed ω0 with respect to distant stars such that the rotation angle is Under these
Polar orbit
We shall now investigate the effect when the light rays move along a circular trajectory passing over the poles. In this case, too, we may take ρ=R=const and ϕ=const. Assuming uniform motion again, we take θ=ω0t. Then, we have, using dρ=0, dϕ=0, and dτ2=0, from metric (5): Assuming that a2/R2⪡1,t=0 when θ=0, we have, on integration, During this time t, the rotating observer describes an
Geodesic orbit
Consider the geodesic motion of the source/receiver having a 4-velocity uμ (≡dxμ/dτ) so that the equations are where Γμνα are the Christoffel symbols formed from metric (5). The problem can be simplified by choosing θ=π/2, so that uθ=0. The geodesic equations this case, do allow such a solution [11]. Then, for a circular geodesic orbit with a constant radius ρ=R, the condition is uρ=0. Then the radial equation turns out to be Defining the
An estimate of the dilatonic charge
In the above, we provided exact formulations of the Sagnac delay for three types of orbits. These lead to correction terms embodied in , , , which reveal, especially, the role of the dilaton parameter ξ. The terms involving only a and M are the Kerr corrections which could be easily evaluated for a given configuration. However, this aside, we could try to have an idea of the possible estimate of the dilatonic charge Q from its contribution to the delay.
One possibility is the following: The
Some remarks
Corrections to the Sagnac delay beyond the basic value δτS could also come from the non-Einsteinian theories such as the Brans–Dicke or other nonminimally coupled scalar tensor theories [7] but, unfortunately, these theories produce only naked singularities instead of black holes. This situation leaves us only with a very few physically viable options in the form of either Kerr–Newman or dilaton gravity. The best Kerr corrections to the Earth bound experiments is of the order of ∼10−15 s and
Acknowledgements
One of us (K.K.N.) wishes to thank the Inter-University Center for Astronomy and Astrophysics (IUCAA), Pune, India for providing several facilities during the work.
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