Propagation of electromagnetic waves through complex space for astronomical redshift investigation

Space has a complex structure and investigations of any electromagnetic wave transmission theory need to consider the inhomogeneous and anisotropic nature. We have selected two cases for our investigations: regions of pulse energy changes and gravitational deflection. Numerical methods have been developed and examples given to show that these conditions do have their localized effects. But, since the total length of those regions are insignificant in comparison with the total transmission distance involved, their inclusion does not significantly alter the linear relationship between wavelength change and distance travelled. The possible exception is the case of gravitational deflection when the waves have passed through densely populated regions of space. Our findings could be of interest to the current debate on Hubble tension.


Introduction
In recent years, we have published three papers [1,2,3] about how the nonlinear Schrodinger equation (NSE) could be used to find out wavelength changes in electromagnetic waves propagating through space.As the only direct and easily measurable physical quantity from space, electromagnetic waves have been widely used to study the universe.For example, redshift, the lengthening of wavelength in starlight, has been used to determine the distance of star, and the size and origin of the universe.To be sure that those applications are reliable, it is important to understand the physics involved in redshift.Up to recent years, there are realizations that we still need new physics to explain this phenomenon [4].Although the physics behind NSE are not new, but we have shown that redshift could be predicted.In this paper, we report our further investigation and provide additional evidence to support the wave propagation theory.
Solving NSE numerically, we have generated waveforms specifically for bright [1], dark [2] and anti-dark [3] solitons.These are respectively white, dark, and grey spectral lines used by astronomers to determine the extent of wavelength changes in starlight coming from space.We have shown that those solitons can propagate stably over the long distance through space.But the previous model used involves only two system parameters: namely, the dispersion coefficient and one for the nonlinear focusing term.In reality, space is a much more complex medium with many regions having localized unique features.In this paper, we shall include two extra features into our model: (i) pulse energy changes due to prevalent atmospheric conditions, and (ii) gravitational deflection of light path.
In propagating through some sections of a vast distance in space, electromagnetic waves could encounter atmosphere conditions that impart or absorb energy from the waves.The two cases of pulse energy changes we consider are (i) elements present in the atmosphere that emit or absorb pulse energy of similar frequency resulting in the increase or decrease of pulse amplitude, and (ii) the CW background that cause pulse energy to change.To account for these additional parameters, NSE has been modified accordingly.
According to General Relativity Theory, travelling light path would be deflected near the proximity of a vast mass.Depending on the system involved, the full mathematical analysis is rather complicated.We shall use an approximate solution [4] that has been derived for cases of small deflections.In this approximate solution, the deflection is found to be inversely proportional to a distance variable.We consider that such an approximation is adequate for us to investigate the contribution of gravitational deflection to redshift.Space should not be modelled simply as an isotropic and homogeneous medium such as quite often been suggested.Photos taken from James Weber Space Telescope (JWST) in its deep space probes [5] have shown clusters of galaxies in certain regions while there are huge empty spaces elsewhere.Along a given light path, a wave will experience different conditions as compared with other paths.The aim of our investigation is to find out whether with pulse energy changes in some sections and gravitational deflections in others, the overall redshift could still be predicted by the wave propagation theory using overall averaged system parameters.Our interest is on the gradient of the wavelength versus distance plot.But our findings are in dimensionless format; we need calibration to convert them into physical entities.
In Section 2 of this paper, we describe how NSE with an external source or with a CW background could be solved numerically.In Section 3, we describe how we calculate the deflection of a stable soliton using the approximate solution of General Relativity Theory.In Section 4, we give numerical examples that provide us with sufficient data for Discussion in Section 5.Although the propagation of solitons is governed by the local conditions that could have varying effects to the propagating wave, those effects are cumulated at each step, and are carried over to the next.It is the overall redshift that is important.This redshift is observed by astronomers in their empirical Hubble law and predicted by astrophysicists in model such as the Standard Model of cosmology.But based on our numerical solutions, we conclude that the electromagnetic wave theory could be used to account for those localized transmission variations investigated.

Stable periodic (SP) bright solitons
The NLS equation for electromagnetic waves (solitons) propagation in dimensionless form is where u is the slowly varying envelope of the axial electric field, and D(x), x, t, and S are the dispersion coefficient, the spatial propagation distance, temporal local time, and external source, respectively.The last term in the left-hand side of Eq. (1) represents self-phase modulation but without a specific system parameter.
To include a CW background, uo into Eq.( 1), let With S = 0, substituting the above into Eq.( 1) to give Using the same numerical procedures as described in our previous papers [1,2,3], Eq. (1) could be solved to give a stable periodic solution along the propagation distance x.The procedures involve the division of the numerical spatial time window of length L into N equal segments.Over each segment, the solution is to be approximated by an economized (M -1) th order power series, Eq. ( 1) is discretised in the t-direction by using collocation points chosen to be the roots of a Chebychev function, Together with the boundary conditions and all the interfacial continuity conditions between any two subdivisions, the set of ordinary differential equations so obtained is in the form, where V is a [M x N] vector consisting of the coefficients of the power series used and A, L, and Q are matrix operators.As Eq. ( 4) is nonlinear, it could be solved by an implicit difference algorithm together with iteration.
For a bright soliton solution, a suitable initial input pulse could be, where L is the length of a given numerical window for t,  an arbitrarily chosen constant and  an adjusting parameter to give a specified pulse energy, E, It is important to set the boundary conditions as The large constant associated with the derivative term will force u to assume a near zero value with zero gradient so that reflection at the boundaries is minimized.
For the stable periodic solution, we integrate Eq. ( 4) to a total distance Z, with the specified dispersion coefficient -D for the first half and D the second half of Z.We use the fact that, for an SP solution propagating through a dispersion map, the input pulse should be similar in shape as the output pulse.To reach this goal, an iterative scheme based on successive halves could be used, where uin , and uout are the input and output pulse to the dispersion map respectively and the superscript i denote the iteration number.It should be noted that SP solitons are special cases of exactly periodic solitons.But, with exactly periodic soliton, the input pulse is exactly the same as the output pulse.
The stable periodic solitons so found could be used as the initial input to start a propagation history through sections that have various system parameters.It should be noted that for those histories the same set of algebraic equations are used with the exception that the steps described in Eq. ( 8) are not included.

Gravitational Deflection
Based on the approximate solution [4] of the General Relativity theory, the light path deflection angle, ∆ϴ is found to be inversely proportional to the distance between the light path and the centre of the mass: where G is the gravitation constant, M is the mass and ∆ is the distance between the wave front and the centre of the mass.Eq. ( 9) could be used in its dimensionless form, where ℰ = 5∆ 23) . We can study the contribution of gravitational deflection by tracking the history using a single arbitrarily chosen parameter C. Numerically, we shall use a new rectangular coordinate system (x1, x2).Tracking the wavefront at a particular step, let the mass be at ((x1)m, (x2)m ) and the wavefront be at ((x1)1, (x2)2); the straight line connecting the wavefront to the centre of the mass is Then, Eq. ( 10) can be used to find the deflection angle ∆, If a wavefront has propagated along a light path making an angle ϴ with the x1-axis and reached ((x1)1, (x2)2), path direction for the next integration step would be along the deflected angle, ϴ + ∆.With the path integration step being ∆x, the change of the wave front position would be ∆x cos (ϴ + ∆) and ∆x sin (ϴ + ∆) in x1-and x2-coordinate, respectively.Knowing the new position, Eqs, ( 11) and ( 10) could be used to find ℰ and ∆, respectively, for the next integration step.

Numerical Examples
For every case of our present investigations, we start with a stable bright solution obtained numerically as described in our previous paper [3].Using L = 40, N = 10, M = 20, ∆x = 0.0005, D = -0.1,E = 0.25, and a dispersion map Z = 6, Figure 1 shows that such a soliton propagates stably with an increasing wavelength.Further more, the same characteristics are observed when it is propagating through a section with a set of different system parameters.We retain the description of periodic because the histories are all repeatable.The pulse width histories found are shown in Fig. 2. The noticeable features are: (i) the gradient is slightly higher for negative uo , and slightly lower for positive uo ; (ii) the gradient in the sections before and after a section with CW background remains the same.The solutions are shown in Fig. 3 in that we can see, in Sections 2 and 4, there are large decreasing and increasing of pulse energy due to the present of uo.But the gradient change between Sections 2 and 3 is very small.The same happens in between Sections 4 and 5. Also shown is the fact that the overall half pulse width change could be accurately predicted by using the average distance weighted D as in Case 4 of Example 1.

but S(x) = s u,
where s = 0.5, -0.5 and 0.5, respectively, in each section.Features of the solution histories are: (i) pulse energy increases or decreases steadily according to the sign of s, and (ii) there is very little change in the wavelength half width gradient in all sections, as can be seen in Figure 4.

Propagation with gravitational deflection
With small deflection, propagation of electromagnetic waves is still governed by Eq. 1.The propagation distance, x, is taking to be along the deflected light path.When integrating Eq. ( 1) by a stepwise procedure, the deflection and tracking of the wavefront are carried out based on procedures described in Section 3.
Example 4 -Using the x1-, and x2-coordinates, the starting position is chosen to be at (0, 0) with the mass located along the straight line x1 = 0.5 Z, where Z is the length of the propagation distance to be investigated.If the line joining the wavefront to the centre of the mass makes an angle, φ, with the x1-axis, the mass is located at (0.5 Z, b), where b = 0.5 Z tan(φ).For this example, the initial φ = 20 o , ϴ = 0, Z = 3, and D = -0.2.For the arbitrarily chosen constant in Eq. ( 10), C = 0.00005 and 0.0001, respectively, for Cases 7 and 8.The solutions are plotted out in Figure 5, The deflection rate (as seen in the direction changes) is the largest at x1 = 0.5 Z, where the wavefront is closest to the mass.Also, a larger C gives larger deflection as expected.Without deflection, the wavefront will move along the x1axis and travel 3 units in that direction.With deflection, the wavefront has travelled the same distance along its path, the same as with the case without deflection, but shorter in term of x1-coodinate.It should be pointed out the scaled down dimensionless length units, x1 and x2, used here are based on the local conditions and would be many orders smaller than x, the propagation distance used in NSE which is in billions of light years.

Discussion
Since the discovery of statistically significant difference in Hubble constant predicted by Big Bang theory-based method and by empirical correlation, this 'Hubble tension' has not yet been resolved.A 2023 review [7] has referenced 531 papers; each has offered one or other scientific solution.But some people would insist on the Big Bang approach often on philosophical belief, while others accept what they consider as more authoritative.Scientists would ask for new evidence such as in the call for new physics [4,7].Although the physics used in Schrödinger equation is not new, it has been widely used, both in theory and in practice, for wave propagation in many diverse fields.However, its application to electromagnetic waves propagating through space is a new suggestion.
There are more than 43 known mechanisms for wavelength changes [8].Most of these mechanisms are localized effects that would not contribute significantly to the overall wavelength changes that is the central issue of Hubble tension.However, if a localized effect is recursive (happening in many sections of the light path), contribution of this type of effect needs to be investigated.We have selected pulse energy change and CW background; both are recursive.
In additional to Hubble tension, another challenge facing Big Bang theory is the fact that a homogeneous and isotropic universe is assumed on a large-scale model and the cosmological principle such that one single Hubble law is supposed to cover all cases no matter how far away they are been observed.But, even on such a scale, the universe is found to be both anisotropic and inhomogeneous [9,10].Based on what we have found numerically, the wave propagation theory needs not rely on the Cosmological principle, although a set of distance averaged parameters could be used to predict the overall wavelength shift.This is due to the fact that the waves are found to be stable, periodical and other innate characteristics of their propagation.
In our numerical investigations of Examples 1 and 2, we have dealt with cases where the pule amplitude has changed gradually together with a small change of the pulse shape due to the present of a CW background.In these cases, as can be seen in Fig. 2, predictions have indicated a small change of the gradient in the redshift-distance plot.But the changes could be cancelled out in the present of sources of opposite sign, as in Case 5 of Example 2 and shown in Fig. 3. Also, the overall wavelength changes can be predicted by using the distance weighted D without CW background.
Pulse energy changes due to an external source proportional to the pulse itself, will only affect the amplitude and not the pulse shape as in Example 3. In these cases, there is no change is the gradient as can be seen in Fig. 4.This is consistent with what we have found previously that pulse energy has not been an important variable in the predictions of wavelength changes [11].
We did not investigate cases of large pulse distortion due to external energy sources.However, if the waves are soliton-like, it is characteristic of such propagating waves to restore to their stable shapes once those external factors no longer exist.
If our interest is the overall wavelength changes over the entire journey through space, it should be pointed out that total contribution due to energy changes would be quite small because of the following reasons: (i) wave length changes is accumulated throughout the entire distance travelled that is measured in billions of light years; the total distance over which the conditions are in favour of energy exchanges could be measured only in millions of light year; (ii) as can be seen in our Examples, there are no dramatic changes in wavelength changes due to energy changes; (iii) wavelength changes could be positive or negative, leading to closer to zero contribution, if the total is to be taken into consideration.
The situation with gravitational deflection is different.Gravity causes the light path to curve.Without deflection, the waves are propagating in a straight line and the distance between two points is always shorter when comparing with a curved light path.In additional, contributions are accumulative over the entire propagating distance through space.Based on our propagation theory and as shown in our numerical Example 4, the wavelength changes over a given time interval is the same with or without deflection.But the apparent gradient in the wavelength change-distance relationship, however, is higher in the present of deflection.The implication is that waves passing through densely populated regions of space would have noticeable longer path length than those passing through sparsely populated regions.Locally, the deflection could be large; Case 8 of Example 4 show a total deflection of nearly 60 o from the original light path.But masses situated at the opposite side of the path could cause negative deflection.The fact that some waves have eventually reached the observers is evidence the nett deflection need not be considered.
Historically, the deflection and lengthening of light path due to gravity had been used to confirm General Relativity Theory.The lengthening of light path had also been used as the scientific argument for an expanding universe in the early days of the Big Bang theory.Although we do not have the actual physical data, in principle, our numerical example is sufficient to demonstrate that light path can be lengthened by gravity.But, if we accept the prediction by General Relativity Theory, this should be the same anywhere in the universe and is not related to the position of the observer.We cannot accept the preposition in Big Bang theory that the rate of wavelength lengthening is proportional to the distance between light source and observers.

Conclusion
(1) Due to the innate nature of soliton, and the balancing of the dispersion term with the nonlinear terms in NSE, electromagnetic waves can propagate stably over a long distance with wavelength changes linearly proportional to distance travelled, although the rate of change is governed by the local prevalent conditions.
(2) Solitons have both wave-like and particle-like transmission characteristics.Due to the latter characteristic, any changes are being accumulated throughout the entire distance travelled.
(3) Since all wavelength changes vary linearly with distance, the distance weighted averages can be used to predict the overall shift that retains the linear relationship over any length of transmission.
(4) Consideration of an inhomogeneous and anisotropic space is needed if localized events are involved.But overall contributions to redshifts observed in starlight are most likely insignificant except for gravitational deflection.

Figure 1
Figure 1 Propagation of bright stable periodic solution

Figure 3 -Example 3 -
Figure 3 -Histories, Cases 4 and 5, of half pulse width and pulse energy histories

Figure 4 .
Figure 4. Propagation histories in the present of an external source

8 Figure 5 .
Figure 5. Propagation in the present of gravitational deflection

Table 1
-Case 5 (See Table2for details) is designed to include two sections, in each, both D and uo have the same values but of opposite sign.In five other sections involved the dispersion coefficient is randomly selected but, for the whole length, the overall distance weighted average is 0.1 that is the same as Case 3 in Example 1.
System parameters used in Example 1 Figure 2. Effects of CW background on the half pulse width histories Example 2

Table 2 -
System parameters for Case 5