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 theory 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 cosmological 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 explanation 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 galaxies 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 obtained using the Hubble Space Telescope near-infrared camera [7]. Figure 3: A 5 GHz radio image of a possible compound 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 equation, g is the ‘out-going’ Green’s function given by [15] exp(ikr) g(r, k) = 4πr which is the solution of

(∇2 + k2)g(r, k) = −δ3(r)

and it is assumed that

u(r, k) = ui(r, k), ∀ r ∈ S

where S is the surface of V and

ui(r, k) = exp(iknˆi · r)

satisfying the homogeneous Helmholtz equation

2 2 (∇ + 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 ∇2 = −δ3(r) (2) 4πr type produced by a spiral galaxy, for example, Let us now assuming that u ∼ ui∀r ∈ 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. (∇2 + 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 ∇ is the Laplacian operator, k = ω/c0 is and because this series scales as k2, k4, k6, ..., for the wavenumber, ω 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) ∃ ∀ r ∈ V where V is a volume of arbitrary shape. The Green’s function trans- formation of equation (1) [14] yields the solution 3 Low Frequency Scattering

u = ui + us If a Helmholtz wavefield oscillates at lower and lower frequencies, then we can consider an asymp- where ui is the incident wavefield, us is the scat- totic solution of the form tered wavefield given by k2 2 u (r, k) = ⊗ γ(r)u (r, k), k → 0. us(r, k) = k g(r, k) ⊗3 γ(r)u(r, k), r =| r | s 4πr 3 i

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

This is a consequence of the fact that the higher where k0 denotes a value for k, k → 0. Consider order terms in the Born series can be ignored a Born scattered Helmholtz waveﬁeld us(r, k) for leaving just the ﬁrst term as k → 0 and because k >> 1 given by

exp(ikr) 1 2 = , k → 0 us(r, k) = k g(r, k) ⊗3 γ(r)ui(r, k) 4πr 4πr giving an exact solution to the problem. We can then write k2 If the incident field is a unit plane wave, then 2 0 us(r, k) = − 2 g(r, k) ⊗3 ui(r, k)[∇ us(r, k0)] k0 u(r, k) = 1 + us(r, k) from which we can derive an expression for the far where field scattering amplitude generated by the field k2 u (r, k) = ⊗ γ(r), k → 0 U0 given by s 4πr 3 s Here, the wavelength of the incident plane wave- k2 u (r, k) = − g(r, k) ⊗ u (r, k)[∇ · U0(r, k )] field is assumed to be significantly larger than the s 2 3 i s 0 k0 spatial extent V of the scatterer. For a given scattering function γ(r) the wavefield is a ‘weak field’ exp(ikr0) r = A(nˆ0, nî), << 1 because of the low values of k required to pro- 4πr0 r0 duce this (asymptotic) result. But this result is where, with ui(r, k) = exp(iknî ·r), writing nˆ0 = r0/ | the general solution to Poisson’s equation r0 | and with ∇2u (r, k) = −k2γ(r) s k2Γ U0 = nÛ 0 = nˆ 0 since, from equation (2), we have s s 4πr2 2 2 2 2 1 2 we obtain ∇ u = ∇ us = k ∇ ⊗3 γ = −k γ. 4πr A(nˆ0, nî) = k2Γ Z nˆ By considering us to be a potential, we can write ˆ ˆ 3 − exp[−ik(n0 − ni) · r]∇ · 2 d r 2 4π r ∇ · Us(r, k) = k γ(r), Us(r, k) = −∇us(r, k). V Integrating over the volume of the scatterer V and Hence, the wavefield us(r, k) (for k >> 1) gener- using the divergence theorem, we can write ated by a scatterer that is simultaneously generating a scattered wavefield u0(r, k ) is, in the far I Z s 0 2 2 3 field (under the Born approximation) determined Us(r, k) · nˆd r = k Γ, Γ = γ(r)d r. by the Fourier transform of the scattering func- S V tion (assuming radial symmetry) f(r) = ∇ · (nˆr−2). If we now consider a scatterer that is a sphere, In other words, the weak field generated by very then the field U will have radial symmetry, i.e. low frequency scattering will diffract a high fre- Us = nÛs. In this case, the surface integral be- quency Helmholtz wavefield, the diffraction pat- 2 comes 4πr Us and we obtain tern (i.e. the far field scattering pattern) being k2Γ determined by f(r). U = , k → 0. s 4πr2 Hence, in the limit as k → 0, Helmholtz scattering 4.1 Diffraction by an Infinitely Thin provides an exact solution for a weak scattered field whose gradient (for the radially symmetric Scatterer case) is characterized by a 1/r2 scaling law. Consider the case where an incident plane wave- 4 Diffraction from a Low Frequency field is travelling in the z-direction, i.e. ui = Scattered Field exp(ikz) and is incident on an infinitely thin scatterer defined by the function γ(r) = γ(x, y)δ(z). The scattered wavefield is then given by For k → 0, us(r, k), which we now denote by 0 us(r, k0), is the solution to p 2 2 2 2 exp(ik x + y + z 2 0 2 us(x, y, z, k) = k p ⊗2 γ ∇ us(r, k0) = −k0γ(r) 4π x2 + y2 + z2

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

0 where ⊗2 denotes the two-dimensional convolu- γ(r), r ∈ V (i.e. us is not of compact support as it tion integral over area S and γ(x, y) ∃ ∀(x, y) ∈ S. is given by the convolution of a function of com- Writing out this result in the form pact support with r−1)? In this case, the scattered ZZ waveﬁeld is given by (under the Born approxima- 2 us(x0, y0, z0, k) = k dxdy tion)

2 2 k 2 0 0 k0 p 2 2 2 us = − g ⊗3 ui∇ u , u = ⊗3 γ. exp[ik (x − x0) + (y − y0) + z0 ] 2 s s γ(x, y) k0 4πr p 2 2 2 4π (x − x0) + (y − y0) + z0 For an infinitely thin scatterer given by γ(x, y)δ(z), it is clear that if the scattered wavefield is now measured in the far field, i.e. for the case when 2 0 k0 x/z << 1 and y/z << 1, then us(x, y, z, k0) = ⊗2 γ(x, y) 0 0 4πpx2 + y2 + z2 1 2 2 2 (x − x0) (y − y0) so that in the (x, y) plane located at z = 0, z0 1 + 2 + 2 z0 z0 2 0 k0 2 2 us(x, y, k0) = p ⊗2 γ(x, y). xx0 yy0 x0 y0 4π x2 + y2 ' z0 − − + + z0 z0 2z0 2z0 and thus, For an incident plane wave ui = exp(ikz), the scattered wavefield us is thus, given by us(x0, y0, z0, k) = exp(ikz ) x2 + y2 u (x, y, z, k) = −k2g(r, k) ⊗ exp(ikz)× 0 exp ik 0 0 A(u, v) s 3 4πz0 2z0 ! ∂2 ∂2 1 where + ⊗2 γ(x, y) . ∂x2 ∂y2 4πpx2 + y2 A(u, v) = k2γ(u, v) = k2F [γ(x, y)] e 2 Repeating the calculation given in the previous ZZ section (for z → 0), the diffracted wavefield now 2 = k exp(−iux) exp(−ivy)γ(x, y)dxdy becomes us(x0, y0, z0, k) = with spatial frequencies u and v being defined by exp(ikz ) x2 + y2 0 exp ik 0 0 A(u, v) kx0 2πx0 ky0 2πy0 u = = and v = = 4πz0 2z0 z λz z λz 0 0 0 0 where 2 Here, F2 denotes the two-dimensional Fourier A(u, v) = −2zk transform operator, the result being the standard " # ∂2 ∂2 1 expression for a diffraction pattern in the far field ×F + ⊗ γ(x, y) . 2 2 2 p 2 or Fraunhofer zone [14] and has been derived in ∂x ∂y 4π x2 + y2 preparation to the following section. Note that although the scatterer is taken to be ‘infinitely thin’ because γ(r) = γ(x, y)δ(z), we still consider the physical thickness of the scatterer to 4.2 Diffraction by an Infinitely Thin be finite, i.e. z 6= 0 (z being taken to be a positive Field real ‘infinitesimal’ for all real k). Now, for an arbitrary function f ↔ fe, where ↔ denotes the In the previous section, we derived the far field transform from real space to Fourier space, diffraction pattern for an infinitely thin scatterer. 2 2 However, suppose this scatterer also radiates a ∂ ∂ 2 2 + f ↔ −(u + v )f,e field generated by low frequency Helmholtz scat- ∂x2 ∂y2 tering from the same scattering function. What 1 2π is the contribution of this field to the diffraction p ↔ √ , of the same incident plane wave within and be- x2 + y2 u2 + v2 yond the extent of the scatterer given that the and we obtain 0 scattered wavefield us is taken to exist within and 2p 2 2 beyond the finite spatial extent of the scatterer A(u, v) = zk u + v γe(u, v).

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

Consider a Gaussian diffactor (a unit amplitude pattern characterised√ by a ring which has a max- Gaussian function with standard deviation σ) imum when r0 = z0λ/( 2πσ) as illustrated in Fig- given by ure 5. Also, observe that the magnitude of the intensity patterns generated by the field ∇2u0 is p s γ(r) = exp(−r2/σ2), r = x2 + y2 significantly less than that generated by scatterer γ, e.g. Figure 5 shows numerical simulations of the I 4z2π2r2 2 = 0 diffraction patterns compounded in the (inten- 2 2 I1 z0 λ sity) functions and only if r0/λ ∼ z/z0 will the magnitude become | γ(u, v) |2 and (u2 + v2) | γ(u, v) |2 of the same order. However, with regard to the e e principal remit of this paper, note that the in- using a two-dimensional Discrete Fourier Trans- tensity generated by the scatterer γ scales as λ−4 2 0 form. The analytical solutions for the intensity whereas the intensity generated by the field ∇ us scales as λ−6.

5 Colour Analysis

If we accept an Einstein ring to be a gravitational diffraction phenomena, then the intensity of the diffracted light scales as λ−6 which explains the colour of the rings (blue light having the short- est wavelength in the visible spectrum). This is analogous to the explanation of why the Earth’s atmosphere is blue in colour. Under the Rayleigh scattering [17] condition in which the wavelength is significantly larger than the physical size of the scatterer (when the Born approximation is valid), the scattering amplitude becomes independent of the scattering angle and the intensity of the scattered field is proportional to λ−4. Thus, the sky is blue, because sunlight is scattered by the electrons of air molecules in the terrestrial atmo- Figure 5: Numerical simulation of the intensity patterns sphere generating blue light preferentially around generated by a Gaussian diffractor with σ = 1.5 for a in all directions [18]. Further, as the Sun ap- 1000 × 1000 regular mesh. Top: Surface plot (left) and proaches the horizon, we have to look more and 2 more diagonally through the Earth’s atmosphere. image (right) for | γe(u, v) | ; Bottom: Surface plot (left) 2 2 2 Our line of sight through the atmosphere is then and image (right) for (u + v ) | γe(u, v) | longer and most of the blue light is scattered out 2 before it reaches us, especially as the Sun gets | us | generated by diffraction from the scatterer γ and diffraction from the field ∇2u0 are by very near the horizon. Relatively more red light s reaches us, accounting for the reddish colour of 4 2 2 2 2 sunsets. π σ 2π σ r0 I1(r0, λ) = 2 4 exp − 2 2 z0 λ λ z0 The λ−6 scaling law associated with gravitational and diffraction provides a method of validating or otherwise the theoretical model presented in this pa- 4π6σ2r2 2π2σ2r2 per. We require a scenario in which the same Ein- I (r , λ) = z2 0 exp − 0 2 0 4 6 2 2 stein ring is recorded simultaneously over a broad z0 λ λ z0 frequency spectrum (e.g. using radio, infrared, respectively. Note that the diffraction for a scat- visible and ultraviolet imaging) in such a way that tering function produces a pattern whose inten- the intensities of each image (relative to a known sity peaks at the centre of the image plane (a stan- source that can be used for calibration) can be dard result in Fourier optics) but that diffraction compared on a quantitative basis. However, data from a low frequency scattered field produces a available to undertake such an analysis are not yet

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

available. Instead, another approach is considered 6 Gravitational Diffraction based on the colour generated by light scattered under different conditions. For Tyndall scattering [19], [20] the intensity of light is proportional to An Einstein ring is an effect that is conventionally λ−2 and for Rayleigh scattering , the light scat- explained in terms of the bending of light through tered intensity is proportional to λ−4. Because of the curvature of space (and time) by a mass. This these wavelength scaling relationships, both Tyn- is a consequence of the field equations for a gravi- dall and Rayleigh scattering generate blue light. tational field. In order to obtain an Einstein ring, However, Tyndall scattering can be expected to the magnitude of the gravitational field must be generate a lighter blue than Rayleigh scattering. relatively high such as that generated by a spi- This is illustrated in Figure 6 which also shows, ral galaxy. Further, in order to generate a near for comparison, the colour of the blue light scat- perfect (complete) ring, the entire galaxy must tered by a gravitational field which is proportion- be well aligned with regard to an observer in the ately darker because of the scaling relationship ‘object plane’. The bending of light by a gravi- characterised by λ−6. This comparison is quanti- tational field has an analogy with the geometrical fied in Figure 7 which shows the differences in interpretation of light interacting with a lens. At the lightness of blue using a Hue, Saturation and the edge of a lens, the light beam is ‘bent’ (dis- Lightness (HSL) colour model. The lightness fac- continuously) by the change in refractive index tor associated with each image is characterised from air to glass and from glass to air - the ex- by the ratios 1:2:3 which is in agreement with the treme edge of a lens acts like a prism. Like an logarithmic scaling ratios 2 ln λ : 4 ln λ : 6 ln λ. optical lens, gravitational ‘lensing’ will produce distortions of the object plane when alignment of the ‘earth-lens-object’ is imperfect.

If we interpret an Einstein ring in terms of the results given in Section 4, then the ring is not due to light being bent (continuously) by the curvature of a space-time continuum but the result of the diffraction of a plane wave (i.e. light) by 2 0 the field ∇ us which is taken to be in the plane of the galaxy and to extend beyond it. This requires the magnitude of the scattering function to Figure 6: Examples of the differences in the lightness (for be very large in order to compensate for z → 0. If a HSL - Hue, Saturation and Lightness - colour model) of we model a (spiral) galaxy in terms of a Gaussian the blue light generated by Tyndall scattering (scattering function, then the ring associated with the diffrac- of light by fine flour suspended in water - left), Rayleigh tion pattern given in Figure 5 is, in this sense, a scattering (scattering of light by the atmosphere - centre) simulation of a complete Einstein ring. The use and ‘gravitational scattering’ (diffraction of light by the of a Gaussian function to model the macroscopic gravitation field generated by a galaxy - right). gravitational field generated by a spiral galaxy is intuitive as the edges of a galaxy will not be dis- continuous (especially on the scale of the wavelength of light!). However, in the case of a black hole, the event horizon defines an edge. In such a case, we might expect gravitational diffraction to produce a number of concentric rings, the black hole being modelled in terms of an opaque disc. Figure 8 shows a double ring pattern which is taken to be generated by the light from three galaxies at distances of 3, 6 and 11 billion light years [21]. The double ring structure is explained Figure 7: The ‘blues’ associated with Tyndall (left), in terms of the light from two distant galaxies Rayleigh (centre) and gravitational (right) light scatter- behind a foreground massive galaxy. The prob- ing obtained by averaging over many images of each ef- lem with this explanation is that there does not fect. appear to be a foreground galaxy in the image! Multiple ring patterns associated with a black

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

Figure 9: Simulation of a double Einstein ring based on the gravitational diﬀraction theory of light. Surface plot (left) and associated grey scale image (right).

7 Discussion

The results developed in this paper encapsulate a phenomenology where the Helmholtz equation is, in effect, being used in an attempt to develop a unified scalar wavefield theory where the wavefield u is taken to exist over a broad range of frequencies. At intermediate frequencies, u is Figure 8: A double Einstein ring pattern taken to be taken to describe waves in the ‘electromagnetic caused by the complex bending of light from two distant spectrum’ and at low frequencies, u is taken to galaxies behind a foreground massive galaxy [21]. describe waves in the ‘gravity wave spectrum’. Low frequency waves (gravity generating waves) are scattered by high frequency waves (matter waves) to produce a gravitational field; intermedi- hole are a prediction of the conventional bending ate frequency waves (electromagnetic spectrum) of light by space-time curvature [22]. The idea are scattered by high frequency waves (e.g. a lens) is that, close to the event horizon, the gravita- but can also be scattered by the field generated tional field is so intense that light can be curved from the scattering of low frequency waves to pro- right around the black hole by 180 degrees or duce gravitational diffraction thus accounting for more to produce a ring associated with the light the blueness of Einstein rings observed in the vis- generated by an object that exists in alignment ible spectrum. with, and behind, the image plane. These multiple Einstein ring predictions are based on argu- In general relativity, the curvature of space-time ments analogous to geometric optics whereas the bends light by the same amount irrespective multiple rings considered here are analogous to of the frequency - there is no dispersion rela- Fourier optics. In this sense, we are interpreting tion. The λ−6 scaling law associated with grav- a gravitational field to be generated by the scat- itational diffraction may be validated (or oth- tering of a long wavelength Helmholtz wavefield, erwise) from appropriate simultaneous observa- 0 i.e. the field Us defines a ‘gravitational field’. Fig- tions of the same Einstein ring (complete or oth- ure 9 shows an example of a simulated double ring erwise) at different wavelengths. Other conse- pattern based on the theory presented in Section quences such as a gravitational field generating 2 2 2 4.2 given by (u + v ) | γe(u, v) | where γ is given by a repulsive force that is proportional to the mass the opaque disc function squared in the relativistic case remain of theoretical consequence only. However, it is noted that ( inflation theory (the expansion of the early uni- 0, r ≤ a; γ(r) = verse) requires gravity to be a repulsive force. 1, otherwise. The model considered leads to the proposition that a gravity field is regenerative and exists for a = 6 computed on a 1000 × 1000 element grid. through the continuous scattering of existing low

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

frequency Helmholtz wavefields. This proposition the result of an object scattering (long wave- may provide an answer to the following question: length) waves that already exist as part of the If nothing can escape the event horizon of a black low frequency component of a universal spectrum hole because nothing can propagate faster than which is, itself, the by-product of the ‘big-bang’. light then how does gravity get out of a black hole? The conventional answer to this question is that the field around a black hole is ‘frozen’ into the surrounding space-time prior to the collapse of the parent star behind the event horizon and 7.2 Compatibility with General Relativ- remains in that state ever after. This implies that ity there is no need for continual regeneration of the external field by causal agents. In other words, the explanation defies causality. In the model pre- The compatibility of this approach with general sented here, the gravitational field generated by relativity can be realised if the wavefield as taken a black hole or any other body is the result of to warp space-time so that space-time is the a causal effect - the scattering of low frequency medium of propagation. Only at very large wave- scalar waves. In this sense, a black hole is just a lengths does the warping of space-time become stronger scatterer than other cosmological bodies so pronounced and over such a large scale that and a gravitational field ‘gets out of a black hole’ Einstein’s field equations can then be used to de- because it was never ‘in the black hole’ to start scribe the physics associated with the geometry with. of the field. In other words, if space-time is taken to be the medium of propagation of all (scalar) wavefields at all frequencies, then the theory of 7.1 Propagative Theories general relativity emerges naturally as k → 0. A two-dimensional and qualitative illustration of Propagative or wave theories of gravity have been this idea is given in Figure 10 which shows four proposed for many years. In 1805, Laplace pro- frames of a simple two-dimensional wave function posed that gravity is a propagative effect and con- as k → 0. It is assumed that the wavefunction sidered a correction to Newton’s law to take into is due to the scattering of a plane wave from a account the observation that gravity has no de- delta function located at the centre of the surface. tectable aberration or propagation delay for its If space is taken to be the medium of propaga- action. Laplace’s ideas were advanced further tion which undergoes curvature as a wave prop- by Weber, Riemann, Gauss and Maxwell in the agates through it then Figure 10 can be taken Nineteenth Century using a variety of ‘corrective to illustrate the curvature of a two-dimensional terms’. In 1898, Gerber, developed a propaga- space into a three dimensional space at increas- tive theory that took into account the perihelion ingly lower frequencies. As k → 0 the wavefield advance of mercury and in 1906 Poincaréshowed is replaced by what appears to be a static curved that the Lorentz transform cancels out gravita- space manifold within the locality of a low fre- tional aberration. After the success of general quency scattering event. The curvature of this relativity (1916) for explaining gravity in terms of manifold is the taken to be responsible for gen- a geometric effect, propagation theories were dis- erating a gravitation force which is attractive in carded. However, more recently, attempts at ex- terms of the influence of one mass upon another plaining gravity in terms of causal effects through and is compounded in terms of Einstein’s field a ‘propagative’ force have been revisited [23] as equation, i.e. debate over the basic Einsteinian postulates (i.e. the invariance of the propagation of light in a vac- 1 8π uum for any observer which amounts to a pre- R − g R + g Λ = T µν 2 µν µν c4 µν sumed absence of any preferred reference frame) has intensified. Moreover, from Laplace to the present, propagation theories of gravity consider where Rµν is the Ricci curvature tensor, R is the an object to be ‘radiating’ a field (in a passive scalar curvature, gµν is the metric tensor, Λ is sense). If general relativity considers gravity to the cosmological constant, G is the gravitational be the result of an object warping space-time, constant, c is the speed of light and Tµν is the then the proposition reported is that gravity is stress-energy tensor [16].

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

Table 1: Table of the equations associated with diﬀerent ﬁeld types.

Equation Field Spin ~s Example Name Type Klein- Scalar s = 0 Higgs boson Gordon Dirac Scalar- s = 1/2 leptons: Spinor electrons, muons Proca- Vector s = 1 m = 0: Figure 10: Qualitative illustration of the function Maxwell photons p −Re[cos(kr)/r], r = x2 + y2 for four frames as k → 0 gluons; (from left to right and from top to bottom). m 6= 0: mesons Rarita- Vector- s = 3/2 None 8 Field and Waveﬁeld Theories of Schwinger Spinor discovered Physics Gravitation Tensor s = 2 gravitons

The field equations for electromagnetic and gravitational fields (i.e. Maxwell’s equations [24] Table 2: The fundamental forces of physics, their ranges and Einstein’s equations [25], respectively) ap- and particle transmission types. pear to have only one thing in common: they both predict wave behaviour (the wavefields be- Force Range Transmitted ing composed of very different ‘fields’ with dif- by Bosons ferent properties), namely, electromagnetic waves Gravitational Long Graviton and gravity waves respectively where, in the lat- m = 0, s = 2 ter case, no direct experimental observations have Electromagnetic Intermediate Photon been made, to date. In quantum mechanics, the m = 0, s = 1 ± quantum fields that are modelled through equa- Weak Short W , Z0 tions such as the Schrödinger[26], Dirac [27], m 6= 0, s = 1 [28], Klein-Gordon (e.g. [29], [30]) and Rarita- Strong Short gluons Schwinger [31] equations, are not fields in the m = 0, s = 1 sense of an electric (vector) field or a gravitational (tensor - a curved vector space) field but wavefields of different types, i.e. scalar (Klein-Gordon Table 1 where m denotes the rest mass and ~ is and Schrödingerequations for the relativistic and the Dirac constant. non-relativistic case, respectively), scalar-spinor (Dirac equations), vector (Proca equations [32], Note that, like the graviton, the Higgs boson is [33]) and vector-spinor (Rarita-Schwinger equa- a hypothetical particle that is taken to explain tions) fields. the origins of mass m which has, to date, not been verified experimentally. The terms ‘Boson’ Apart from the Schrödingerequation, all of the and ‘Fermion’ relate to the fact that the statis- equations listed above describe relativistic quan- tical behaviour of integer spin particles can be tum fields. They are all ‘products’ of the fact that, classified in terms of Bose-Einstein statistics and given the postulates of quantum mechanics, Ein- half-integer spin particles, in terms of Fermi-Dirac stein’s special theory of relativity allows for the statistics. existence of scalar, scalar-spinor, vector, vector- spinor and tensor fields. In each case, the field, as Vector bosons are considered to mediate three characterised by a given operator, is taken to de- of the four fundamental interactions in ‘particle’ scribe a ‘particle’ (a localised entity) that is clas- physics, i.e. electromagnetic, weak and strong in- sified in terms of a Boson or Fermion which have teractions, and tensor bosons (gravitons) are as- integer or half-integer spin (the intrinsic angular sumed to mediate the gravitational force as sum- momentum) respectively. This is compounded in marised in Table 2.

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

Of the four fundamental forces in nature, gravity of charged black holes, the existence of magnetic was the first to be ‘invented’ but, to this day, monopoles and superluminal (faster than light) remains the most elusive. particles (Tachyons) with an imaginary mass that can be described by a Proca field with a negative In electromagnetism and general relativity, the square mass [36], [37] and [38]. field equations are considered to be fundamental, the wave properties of these fields being a The principle associated with deriving the Proca consequence of decoupling (under certain condi- equations can be applied to other field equations tions) the field equations. In other words, the such as the Einstein equations for a gravitational wave properties of these fields are, in a sense, a field. The Proca-Einstein equations have been by-product of writing a set of coupled equations used as a basis for modelling the interaction of in terms of a single or set of equations of the gravitational fields with dark matter, for exam- same (wave) type. What if a wave equation was ple [39]. In string theory, there is tentative ev- to determine the form of the field equations and idence that non-Riemannian models such as the thus the characteristics of the field(s)? The first Einstein-Proca-Wyle equations may account for to consider such an approach was the Romanian dark matter [40]. However, in the context of this born Alexandru Proca who derived the Proca or paper, the Proca equations are an example of the Proca-Maxwell equations. modification and extension of a set of field equations in order that a given wave equation is sat- Given that Maxwell’s equations can be decou- isfied. Thus, in the derivation of the Proca equa- pled to produce inhomogeneous wave equations tions, the Klein-Gordon wavefield is the governing for the electric scalar potential φ and the mag- function and not the electric and magnetic fields. netic potential vector A, Proca’s idea was to mod- In other words, the Proca equations are based on ify Maxwell’s equations in order to produce inho- ‘tailoring’ a field to ‘fit’ a wavefield. This leads mogeneous Klein-Gordon equations for φ and A us to consider an approach in which unification is given by attempted, not in terms of a unified field theory but in terms of a unified wavefield theory where 1 ∂2 ρ 2 2 a wavefield is not just the governing function but ∇ − 2 2 φ(r, t) − κ φ = − c0 ∂t 0 the governing principle. and If a unified field theory (unifying gravity and elec- 2 tromagnetism, for example) were available, then, 2 1 ∂ 2 ∇ − A(r, t) − κ A = −µ0j by induction, we might expect that the unify- c2 ∂t2 0 ing field equations yield a unifying wave equation. respectively where 0 is the permittivity, µ0 is the Since a unified field theory is not currently avail- permiability, ρ is the charge density, j is the cur- able, our approach is to attempt to construct a unified wavefield theory in which a field is the rent density and κ = mc0/~. These equations im- ply that φ and A and thus, the electric and mag- product of certain characteristics of a wavefield. netic fields are effected by mass. Thus, the basic idea is to develop a universal physical model that is based on a wavefield equa- The Proca equations are relativistic field equation alone and attempt to explain the character- tions that describe massive electromagnetic fields istics of a field from the wavefield. In this pa- or massive photons (spin 1 vector bosons). They per, we have adopted the (inhomogeneous) scalar form the foundations for the electro-weak the- Helmholtz equation as a governing equation. We ory (the unification of electromagnetism with the have then considered the case when the wave- ‘weak’ force) where it is assumed that the electro- length approaches infinity and used the result to magnetic fields of the early universe had signifi- model the ‘diffraction’ of light by a field that we cantly greater (relativistic) energies than now, i.e. interpret to be a gravitational field, thereby ex- the electromagnetic and the weak force are mani- plaining why an Einstsin ring is blue. festation of the same force at relativistic energies. ± Vector Bosons (W and Z0 bosons) are taken to be mediators of the weak interaction. However, the 9 Concluding Remarks Proca equations, as a description for massive photons, have a number of other implications. These Any propagation/scattering theory of gravity include variations in light speed, the possibility must address some basic known observations:

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

• Gravity has no detectable aberration or prop- is the result of an object scattering long wave- agation delay for its action leading to effects length waves. The compatibility of this idea with predicted by general relativity such a gravito- general relativity can be realised if the wavefield magnetism; as taken to warp space-time so that space-time is the medium of propagation. Only at very large • the finite propagation of light causes radiation wavelengths does the warping of space-time be- pressure for which gravity has no counterpart come so pronounced and over such a large scale pressure. that Einstein’s field equations can then be used to describe the ‘geometry’ of the field. In other These results represent the most vital evidence words, if space-time is taken to be the medium of with regard to gravity being a geometric and not propagation of all (scalar) wavefields at all fre- a propagative effect. For example, in an eclipse quencies, then Einstein’s equations ‘emerge’ as of the Sun, the gravitational pull on the earth by k → 0. A two-dimensional and qualitative illus- this 3-body (Sun-Moon-Earth) configuration in- tration of this idea is given in Figure 10 which creases. By comparing the delay in time it takes shows four frames of a simple two-dimensional to observe the visible maximum eclipse on Earth wave function as k → 0. It is assumed that the (which can be calculated from knowledge of the wavefunction is due to the scattering of a plane distance of the Moon from the Earth) with the wave from a two-dimensional delta function lo- equivalent gravitational maximum, then if grav- cated at the centre of the surface. If space is ity is a propagating force, it appears to propa- taken to be the medium of propagation which un- gates at least 20 times faster than light! [41] Ir- dergoes curvature as a wave propagates through it respective of whether this value is valid or not, a then Figure 10 can be taken to illustrate the cur- fundamental issue remains which is compounded vature of a two-dimensional space into a three di- in the question: what is the speed of gravity? If mensional space at increasingly lower frequencies. we consider gravity to be a low frequency weak As k → 0 the wavefield is replaced by what ap- scattering effect, then in order to account for the pears to be a static curved space manifold within lack of propagation delay, it must be assumed that the locality of a low frequency scattering event. the speed of gravity is greater than the speed of The curvature of this manifold is the taken to light. This is contrary to the Einsteinian postu- be responsible for generating a apparent gravi- lates if these postulates are taken to apply to all tational force. This idea is validated through the wavefields irrespective of their wavelength. The explanation as to why an Einstein ring is blue, a model presented here assumes that the speed of by-product of the proposition that gravity is a low gravity is the same as the speed of light c0. How- frequency scattering effect in which space-time is ever, the asymptotic result k → 0 used to define the medium of propagation and scattering. a gravitational field yields, what will appears to be, an instantaneous effect from a wavefield that Acknowledgments is taken to propagate at the speed of light. The wavelength is so long compared to the distances The author is supported by the Science Founda- associated with a Sun-Moon-Earth system, for ex- tion Ireland Stokes Professorship Programme and ample, that the speed of gravity will appear to be is grateful to Professor Michael Rycroft for his en- 0 significantly faster than the speed of light (i.e. Us couragement and advice with regard to the pub- is observed to be an instantaneous field). lication of this work. In general relativity, the curvature of space-time bends light by the same amount irrespective of References the frequency, i.e. there is no dispersion relation. [1] Voit, M., Hubble Space Telescope: New Views of This is the case for a medium that is a perfect vac- the Universe, H. N. Abrams, Inc., 2000. uum although it is possible to extend the theory of relativity to include the case of linear but non- [2] DeVorkin D. H., Hubble: Imaging Space and homogeneous electrically and magnetically polar- Time National Geographic Society, 2008. ized media [42] [3] HubbleSite News Release Number: STScI- If general relativity considers gravity to be the 2005-32, Hubble, Sloan Quadruple Number of result of an object warping space-time, then the Known Optical Einstein Rings, November 17, proposition reported in this paper is that gravity 2005;

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

http://hubblesite.org/newscenter/archive/ [16] Einstein A., “The foundation of the general releases/2005/32/image/a/ theory of relativity”, Annalen der Physik, IV 29, pp. 779-822, 1916 [4] Cabanac, R. A., Valls-Gabaud, D., Jaunsen, A. O., Lidman, C. and Jerjen, H., “Discovery [17] Young, A. T., “Rayleigh scattering”, Physics of a High-Redshift Einstein Ring”, Astronomy Today, Vol. 35, No. 1, pp. 28, 1982. and Astrophysics, Vol. 436, No. 2, pp. L21 - L25, 2005. [18] Bohren, C., Selected Papers on Scattering in the Atmosphere, SPIE Optical Engineering Press, [5] Kochanek1 C. S., Keeton, C. R. and McLeod, Bellingham, 1989. B. A., “The Importance of Einstein Rings”, The Astrophysical Journal, Vol. 547, No. 1, pp. [19] Bohren, C. F. and Huﬀmann, D. R., Ab- 5059, 2001 sorption and Scattering of Light by Small Particles, New York, Wiley-Interscience, 2010. [6] Robberto, M. and Sivaramakrishnan, A., Bacinski, J. J., Calzetti, D. Krist, J. E., [20] Mishchenko, M., Travis, L. and Lacis, A. Scat- MacKenty, J.W., Piquero, J. and Stiavelli, tering, Absorption, and Emission of Light by Small M. (2000). “The Performance of HST as an Particles, Cambridge University Press, 2002. Infrared Telescope”, Proc. SPIE 4013, pp. 386393, 2000. [21] Science News, Hubble Finds Double Einstein Ring, 2008 [7] Astronomy Picture of the Day, A Bulls-Eye http://www.sciencedaily.com/releases/ Einstein Ring, March 30, 1998; 2008/01/080110102319.htm http://apod.nasa.gov/apod/ap980330.html [22] Werner, M. C., An, J. and Evans, N. W., “On [8] MERLIN/VLBI National Facility Multiple Einstein Rings”, Royal Astronomical http://www.merlin.ac.uk/ Society, Vol. 391, Issue 2, pp. 668674, 2008

[9] Astronomy picture of the day, sdssj1430: A [23] Van Flandern T. C., Dark Matter, Missing Plan- galaxy Einstein ring, 28.07.2008; ets and New Comets: Paradoxes Resolved, Origins http://www.astronet.ru/db/xware/msg/ Illuminated, North Atlantic Books, Berkeley, 1228939 1993. [10] Ellis R. S., “Gravitational lensing: A unique [24] J. C. Maxwell, “A Dynamical Theory of the probe of dark matter and dark energy”, Philo- Electromagnetic Field”, Philosophical Trans- sophical Transactions of the Royal Society A, Vol. actions of the Royal Society of London 155, 368, pp. 967-987, 2010 459-512, 1865. [11] Einstein, A., “Lens-like Action of a Star by [25] A. Einstein, “The Foundation of the General the Deviation of Light in the Gravitational Theory of Relativity”, Annalen der Physik, Field”. Science 84 (2188), pp. 506507, 1936. IV, Floge 49, 770-822, 1916. [12] Renn, J., Sauer, T. and Stachel, J., “The Ori- http://www.alberteinstein.info/gallery/ gin of Gravitational Lensing: A Postscript to gtext3.html Einstein’s 1936 Science paper”, Science 275 (5297), pp. 184186, 1997. [26] E. Schr¨odinger,“Quantization as an Eigen- value Problem”, Annalen der Physik, 489, [13] Dominik, M. “Studying Planet Populations 79, 1926. with Einstein’s Blip”, Philosophical Transac- tions of the Royal Society A, Vol. 368, pp. 3535- [27] P. A. M. Dirac, “The quantum theory of the 3550, 2010. electron” Proc. R. Soc. (London) A, 117, 610- 612, 1928. [14] Blackledge J. M., Digital Image Processing, Hor- wood Publishing Limited, 2006. [28] P. A. M. Dirac, “The quantum theory of the electron: Part II” Proc. R. Soc. (London) A, [15] Blackledge J. M., Electromagnetic Scattering So- 118, 351-361, 1928. lutions for Digital Signal Processing, PhD Thesis: Studies in Computer Series 108, University [29] J. J. Sakurai, Advanced Quantum Mechanics, Ad- of Jyvaskyla, 2010. dison Wesley, 1967, ISBN: 0-201-06710-2.

(Advance online publication: 24 August 2011)

IAENG International Journal of Applied Mathematics, 41:3, IJAM_41_3_02 ______

[30] A. S. Davydov, Quantum Mechanics (2nd Edi- tion), Pergamon, 1976, ISBN: 0-08-020437-6. [31] W. Rarita and J. Schwinger, “On a Theory of Particles with Half-Integral Spin”, Phys. Rev. 60, 61-61, 1941. [32] A. Proca, “Fundamental Equations of Ele- mentary Particles”, C. R. Acad. Sci. Paris, 202, 1490, 1936. [33] W. Greiner, Relativistic Quantum Mechanics (3rd edition), Springer, 2000. [34] J. A. Stratton, Electromagnetic Theory, McGraw-Hill, 1941. [35] D. J. Griffiths, Introduction to Quantum Mechan- ics (Second Edition), Prentice Hall, 2004 . [36] R. Tomaschitz, “Einstein Coefficients and Equilibrium Formalism for Tachyon Radia- tion”, Physica A, 293, 247-272, 2001. [37] R. Tomaschitz, “Tachyon Synchrotron Radi- ation”, Physica A, 335, 577-610, 2004. [38] R. Tomaschitz, “Quantum Tachyons”, Eur. Phys. J. D32, 241-255, 2005. [39] X. Bei, C. Shi and Z. Liu, “Proca Effect in Kerr-Newman Metric”, Int. J. of Theoretical Physics, 43, 1555-1560, 2004. [40] R. Scipioni, “Isomorphism Between Non- Riemannian Gravity and Einstein-Proca- Wyle Theories Extended to a Class of Scalar Gravity Theories”, Class. Quantum Gravity, 16, 2471-2478, 1999. [41] T. C. Van Flandern, “The speed of gravity: What the experiments say”, Physics Letters A, 250, pp. 1-11, 1998. [42] A. Nicolaide, “Electromagnetic Energy- Momentum Tensor for Non-Homogeneous Media in the Theory of Relativity”, IAENG International Journal of Applied Mathemat- ics, 39:4, pp. 253-264, 2009.

(Advance online publication: 24 August 2011)