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Why Is an Einstein Ring Blue?

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Why Is an Einstein Ring Blue? IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______________________________________________________________________________________ Why is an Einstein Ring Blue? Jonathan M Blackledge ∗ Abstract|Albert Einstein predicted the existence of `Einstein rings' as a consequence of his general the- ory of relativity. The phenomenon is a direct result of the idea that if a mass warps space-time then light (and other electromagnetic waves) will be `lensed' by the strong gravitational field produced by a large cos- mological body such as a galaxy. Since 1998, when the first complete Einstein ring was observed, many more complete or partially complete Einstein rings have been observed in the radio and infrared spectra, for example, and by the Hubble Space Telescope in the optical spectrum. However, in the latter case, it is observed that the rings are blue providing the light Figure 1: A gallery of Einstein rings in the optical spec- is not red shifted. The gravitational lensing equation trum obtained using the Hubble Space Telescope [3]. does not include dispersion (i.e. wavelength depen- dent effects) and thus, can not account for this `blue shift' and, to date, there has been no satisfactory ex- planation for this colour phenomenon. In this paper in the infrared [5] or radio spectra, where false we provide an explanation for why Einstein rings are blue using a linear systems theory approach based colour coding is used for image display purposes on the idea that a gravitational field is generated by only, i.e. the colours are of limited physical sig- the scattering of very low frequency scalar waves in nificance [6]. For example, Figure 2 show an ex- which the medium of propagation is space-time and ample of a complete Einstein ring taken using the that light waves can be both bent and diffracted by Hubble Space Telescope near-infrared camera [7] this field. The latter effect provides a quantitative and Figure 3 shows a 5 GHz radio image of a result that explains why an Einstein ring is blue. compound lensing effect produced by two galax- ies where neither the lensing galaxy nor the lensed Keywords: Einstein ring, Image analysis, Simulation object(s) has yet been identified optically [8]. and Modelling, Gravity Although there have been some minor comments 1 Introduction with regard to the blueness of Einstein rings, they are clearly inadequate with regard to a phys- The Hubble Space Telescope has provided a ical explanation of the phenomenon. For ex- wealth of information of cosmological significance ample, the Astronomy Picture of the Day, July since it came into operation in 1993 [1], [2]. This 28, 2008 given in Figure 4, is accompanied by has included optical images of Einstein rings, the following text [9]: What's large and blue and some typical examples being provided in Figure 1 can wrap itself around an entire galaxy? A gravitation [3]. While the existence of such rings is predicted lens image. Pictured above on the left, the gravity of by Einstein's general theory of relativity, to date, a normal white galaxy has gravitationally distorted the no reasonable explanation has been given as to light from a much more distant blue galaxy. In [10] why, apart from some `red-shifted' cases [4] ob- the phrase `...strongly lens background blue star- served in the far-field (i.e. billions of light years forming galaxies...' is referred to as the cause for away), they are blue as shown in Figure 1. This the colour of the ring. It is clearly not conceiv- observation is in contrast to images of Einstein able that all Einstein rings observed to date (in rings that are not in the optical spectrum but the visible spectrum) are the coincidental result of gravitationally lensed light from `blue galax- ∗Dublin Institute of Technology, Kevin Street, Dublin 8, Ire- land; http://eleceng.dit.ie/blackledge; [email protected]; ies' or `blue star-forming galaxies'. Galaxies are +35 31 402 4707. certainly not blue, but rather, cosmological bod- (Advance online publication: 24 August 2011) IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______________________________________________________________________________________ Figure 2: A near-infrared image of an Einstein rings ob- tained using the Hubble Space Telescope near-infrared camera [7]. Figure 3: A 5 GHz radio image of a possible com- pound gravitational lens obtained using the MERlIN ies that radiate electromagnetic waves over a vast (Multi-Element Radio-Linked Interferometer Network) spectrum. radio telescope [8]. Einstien rings were first predicted by Albert Ein- stein in 1936 [11] and although he originally alignment of the source, lens and observer. This judged that `there is no hope of observing this results in a symmetry around the lens, causing phenomenon directly', the phenomena is now a a ring-like structure with an `Einstein Radius' in relatively routine observation. There are many radians, given by [12] examples of Einstein rings with diameters up s to an arcsecond as shown in Figure 1. How- 4GM DLdS θ = 2 ever, because the mass distribution of the lenses c0 DLS is not perfectly axially symmetrical, or because the source, lens and observer are not perfectly where G is the gravitational constant M is the aligned, many of the observation are of imper- mass (of the lensing object) DL is the angular di- fect or partial Einstein rings. The study of Ein- ameter distance to the lens, DS is the angular stein rings has become an important aspect of cos- diameter distance to the source, DLS is the an- mology in general. This includes the bending of gular diameter distance between the lens and the starlight by the gravity of intervening foreground source and DLS 6= DS − DL over cosmological dis- stars which is now commonly referred to as `gravi- tances in general. The angular diameter distance tational microlensing and has become a technique to a cosmological object is defined in terms of an to detect planets orbiting stars [13]. object's actual size divided by the angular size of the object as viewed from earth. This result is in- Einstein rings are caused by the gravitational dicative of the theory of general relativity being lensing which in turn is a consequence of Ein- a `geometric' interpretation of gravity which does stein's theory of General relativity. A fundamen- not include effects that depend on the wavelength tal consequence of this theory is that mass dis- of the light that is bent. Consequently, general torts space-time and thus, instead of a ray light relativity is not able to explain why an Einstein from a source travelling in a straight line through ring is blue. This paper attempts to explain the a three-dimensional space, it is bent as a result of colour phenomena using a linear systems theory the distortion generated by the presence of a mas- approach based on scalar Helmholtz scattering to sive body. An Einstein Ring is a special case of evaluate the effect of light being not just bent but gravitational lensing which is caused by the exact ‘diffracted’ from a `thin' gravitational field of the (Advance online publication: 24 August 2011) IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______________________________________________________________________________________ and ⊗3 denotes the three dimensional convolution integral over r. With regard to this integral equa- tion, g is the `out-going' Green's function given by [15] exp(ikr) g(r; k) = 4πr which is the solution of (r2 + k2)g(r; k) = −δ3(r) and it is assumed that u(r; k) = ui(r; k); 8 r 2 S where S is the surface of V and ui(r; k) = exp(ikn^i · r) satisfying the homogeneous Helmholtz equation 2 2 (r + k )ui(r; k) = 0 Note that 1 g(r; k) = ; k ! 0 4πr Figure 4: A near complete Einstein ring showing the dis- and thus, tinctive blueness of the phenomenon [9]. 1 r2 = −δ3(r) (2) 4πr type produced by a spiral galaxy, for example, Let us now assuming that u ∼ ui8r 2 V so that the where the gravitational field is taken to be due scattered field is given by to the scattering of low frequency scalar waves. 2 The diffraction effect generates an expression for us(r0; k) = k g(r; k) ⊗3 γ(r)ui(r; k) the intensity of an Einstein ring that depends on −6 This assumption provides an approximate solu- the wavelength of light according to a λ power tion (the Born approximation) for the scattered law where λ is the wavelength. The paper then field which is valid if k2kγ(r)k << 1. The result explores the consequences of this result in terms can be considered to be a first approximation to of a `wavefield theory' of gravity. the (Born) series solution given by 2 2 Helmholtz Scattering us(r; k) = ui(r; k) + k g(r; k) ⊗3 γ(r)ui(r; k) 4 The three-dimensional inhomogeneous Helmholtz +k g(r; k) ⊗3 γ(r)[g(r) ⊗3 γ(r)ui(r; k)] + ::: equation for a scalar wavefield u is given by [14] which is valid under the condition k2kγ(r)k < 1. (r2 + k2)u(r; k) = −k2γ(r)u(r; k) (1) Each term in this series expresses the effects due to single, double and triple etc. scattering events 2 where r is the Laplacian operator, k = !=c0 is and because this series scales as k2, k4, k6, ..., for the wavenumber, ! is the angular frequency. Con- k << 1 (i.e. low frequency wavefields), the Born sider a scattering function γ which is of compact approximation becomes an exact solution. support, i.e. γ(r) 9 8 r 2 V where V is a volume of arbitrary shape.
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