Signature of Gravitational Waves in Stellar Spectroscopy

The possible detection of gravitational waves by interferometric observations of distant light sources is studied. It is shown that a gravitational wave affects the interferometric pattern of stellar light in a particular way. Michelson and Hanbury Brown-Twiss interferometers are considered, and it is shown that the latter is the most adequate for such a detection.


Introduction
A gravitational wave (GW) could be detected indirectly by its interaction with the light emitted by astronomical objects.Thus, for instance, the passage of a GW produces a time delay in the signal received from distant sources (Estabrook and Wahlquist [1]).Similarly, the presence of a stochastic background of GWs can be inferred from a statistical analysis of pulsar timing (Hellings and Downs [2]).GWs can also interact with the polarization of electromagnetic waves (Hacyan [3] [4]).
In this paper, we study the effect of GWs on the interferometry of stellar light.Two basic types of interferometric devices used in astronomy are considered: the Michelson (see, e.g., [5]) and the Hanbury Brown-Twiss [6] interferometers.The former uses the interference between two signals, and the latter uses the interference between intensities of light.An intensity interferometer has, in general, some advantages over a Michelson interferometer.It will be shown in the following that the passage of a GW could be more easily detected by intensity interferometry.
Section 2 of the present paper is devoted to the analysis of an electromagnetic wave in the presence of a plane fronted GW.The analysis is based on previous works (Hacyan [3] [4]) in which the form of the electromagnetic field is deduced using a short-wave length approximation.A general formula for the correlation of electric fields is obtained and the result is applied to interferometric analysis in Section 3; particular cases are worked out.

The Electromagnetic Field
The metric of a plane GW in the weak field limit is where the two degrees of polarization of the GW are given by the potentials ( ) f u and ( ) g u , which are functions of u only.The relation with Minkowski coordinates t and z is ( ) ( ) In the following, quadratic and higher order terms in f and g are neglected, and we set 1 c = .The direction of a light ray in the absence of a GW is k, with k ω = , the frequency of the (monochromatic) wave.We set ( ) sin cos , sin sin , cos , thus defining the angles θ and φ .In the following, it will be convenient to define the functions ( ) ( ) ( ) In the short-wave length approximation, the electromagnetic potential is taken as e , ≡ is a null-vector defining the direction of propagation of the electromagnetic wave, and a α is a four-vector such that 0 a K µ µ = .
The electromagnetic vector is [4] ( ) e , iS E i a a t K where t α is a time-like four-vector and K t µ µ Ω = − is the frequency measured by a detector with t α tangent to its world-line.Choosing and the eikonal function is As in Ref. [4], for a plane wave we use a gauge such that 0 v a = , which is equivalent to where gw n is the unit vector in the direction of propagation of the GW.
The four vector a α depends on the coordinate u through the functions ( ) f u and ( ) where x a and y a are constants defining an electromagnetic plane wave in the absence of GWs.Let us use a tetrad ( )  ( ) ) Accordingly the tetrad components of a α and K α are , , , , ( ) Notice in particular that ( ) , as it should be.
The electric field in tetrad components is e , and of course ( )

Correlations
For an electromagnetic plane wave with wave vector ( ) K u α , we find after some lengthy but straightforward algebra (keeping only terms of first order)  for circular polarizations).

Interferometry
Consider two detectors with space-time coordinates 1 x and 2 x , each receiving two plane electromagnetic waves with wave-vectors 1 k and 2 k , and use the shorthand notation where ( ) ( ) ( 1 cos ; , the subindexes a, b and j refer to the labels 1 and 2 of x and k.
A Hanbury Brown-Twiss interferometer permits to measure the interference between intensities: 1;1 1;1 1;2 1;2 2;1 2;1 2;2 2;2 † † † † 1;1 1;2 2;2 2;1 1;2 1;1 2;1 2;2 , where the second term is the interference between the two intensities.Define and for a Hanbury Brown-Twiss interferometer: Define also the complex functions ) In the absence of GWs, 0 ± ℜ = , and implying that 1 2 I I is time independent.It thus follows that the time variation of 1 2 I I is due entirely to the presence of a GW.This time dependence can be made explicit setting ( ) , and ± ∆ℑ are small terms due to the GW.This implies that the terms ± ℜ and ± ∆ℑ are of first order in the potentials f and g of the GW.
It should be noticed that the field correlation I contains terms such as ( ) , which are highly oscillatory and hinder a precise measurement with a Michelson interferometer.On the other hand, such terms do not appear in the correlation of the intensities: The time dependence is included only in the terms + ℜ and − ∆ℑ , which are entirely due to the passage of the GW.The term with ( ) x k k ∆ ⋅ + is not present in this last formula.

Temporal Coherence
As a particular application of the above formulas, we can calculate the temporal coherence of a single signal in the presence of a GW.This can be obtained setting which is the only relevant term for the time correlation of the intensity correlation, and is entirely due to the GW.

Sinusoidal Waves and Pulses
In the particular case of a sinusoidal monochromatic GW of frequency gw ω , we can set where 0 0 e i h h α ≡ is a complex constant and α a constant phase.
With this notation, we have for a Michelson interferometer: