Conventional Modern Physics Theories, especially Maxwell’s [1] Electrodynamics (MED), Einstein’s [2] Special Theory of Relativity (STR), Quantum Electrodynamics (QED) and the Standard Model (SM) of Particle Physics are all based on the seemingly sacrosanct idea that the speed [ c 0 = 2.99792458 × 10 8   m ⋅ s − 1 (CODATA 2018)¹] of Light in vacuo is an inviolable Fundamental Constant of Nature. This constancy of the speed of Light is assumed to hold for all electromagnetic waves, down from the weakest radio wave to the most energetic γ-rays. In these models (MED, QED, SM etc.), Photons are assumed to be massless—i.e., their rest mass, m₀, is taken to be identically equal to zero. Whether or not the speed of Light is a constant across the entire electromagnetic spectrum plays a fundamental role in All of Physics. If Photons have a non-zero rest mass—no matter how small this mass may be—for so long as it is not identically equal to zero, various Key Theories of Modern Physics will be affected drastically. Despite the fact that enormous successes has been achieved based upon the theories aforementioned, it is still necessary to put this assumption of a massless Photon to the test using as many independent methods as is possible—of which—the time delay in γ-Ray bursts is one of the scenarios were this idea of massive Photon can be put to the test.

For example—in recent times, astrophysical phenomenon such as the time delays observed in Gamma Ray Bursts (GRBs) (see e.g., Refs. [4]-[14]) and Fast Radio Bursts (FRBs) (see e.g., Refs. [7][15][16][17][18]) have brought to the fore of physics this very idea that the Photon may not be massless as we have long assumed. In these time delays, it is observed that Photons of different frequencies supposedly emanating from the same GRB and FRB event arrive at the telescope at different times. Surely, if these Photons are coming from the same GRB/FRB-event where they were supposedly released simultaneously, it is naturally expected that these Photons should arrive at the telescope at the same time if they are travelling along the same path at the same speed as we assume them to be doing. The reality is that, rather surprisingly, Photons of different frequencies arrive at the telescope at different times, with the higher frequency Photons systematically and consistently arriving first. The one common and popular possibility that has been and continues to be explored for this unusual occurrence is that—Photons may very well be endowed with a non-vanishing mass. In Section 3, we discuss this model used to explain how massive Photons should lead to time delays and this being a result of Photons having a non-zero mass.

While we are of the strongly view (see e.g., Refs. [19][20][21][22]) that the Photon may very well be massive, at present, we do not think that this is the reason for the time delays observed in GRBs. For example, we know that the Universe is richly endowed with stars that prodigiously pour out their stellar winds into the interstellar space. Like the Solar wind, this stellar wind of stars should comprise equal portions of fast moving Electrons and Protons. Apart from the Electrons and Protons, there should also be in this stellar wind, a number of unstable charged subatomic particles e.g. Pions ( π + , π − ) , Mouns ( μ + , μ − ) , Tauons ( τ + , τ − ) , etc. These stable fast moving Electrons and Protons must—somehow—be smoothly smeared out uniformly, homogeneously and isotropically across all of space. As such, it is logical and reasonable to assume that at the very least, interstellar space must be a rarefied plasma medium.

If we are to accept the above stated assumption—and—knowing very well that the speed of Light in a non-vacuo medium such as a plasma is going to be different from its vacuo value of unit—and—further knowing that, this speed will not only be different from the vacuo value, but will depend on the particular frequency of the Photon, it follows from this—that, it is possible that the time delays observed in GRBs may very well be due to these rays propagating in a rarefied cosmic plasma. Hence, in this article, we explore this possibility that time delays observed in GRBs may be a result of these electromagnetic waves travelling in a rarefied plasma medium.

First: as already said, in GRBs events—it has been observed that γ-rays of different energies emanating from the same event arrive at the telescope at different times where these γ-rays are supposed to propagate at the same speed of Light: v g = c 0 / n , for the given medium whose refractive index is n. These GRBs were serendipitously discovered (during the so-called Cold War in the 1960s by US VELA spy satellites) and first reported by Klebesade and Olsen [23]. Furthermore, these GRBs seem to hold potent seeds to probe Lorentz invariance via the observed time delays in the arrival times of γ-rays of different energies from these GRBs events. Lorentz invariance is a very important fundamental symmetry in physics and its violation—if confirmed by experiments—can have serious reverberations across all Disciplines of Physics.

Second: as already aforementioned, we also have the phenomenon of FRBs, and these FRBs are one of the newest and latest discoveries in the World of Astronomy. In a nutshell, a FRB is a high-energy astrophysical phenomenon of unknown origin manifested as a transient radio pulse lasting a few milliseconds on average and the first of such was discovered by pulsar astronomers—Duncan Lorimer and his student David Narkevic in 2007; while they were looking through archival pulsar survey data [18], and, for this reason, a FRB is sometimes referred to as a Lorimer Burst [24]. While FRBs are predominately assumed to be of extragalactic origin, their exact origins and cause is uncertain.

Before we close this introductory section, we must hasten to say that the present article is the first in a four paper series.

1) In the present [Paper (I)], we consider four GRB time delays between radio Photons pairs that gave a reasonable good correlation.

2) In the second part [hereafter, Paper (II)], we reconsider the fitting procedure here applied, whereby we improve on the apparent groupings of the GRB time delays.

3) In the third part [hereafter, Paper (III)], we consider yet another set of four different GRB time delays that do not give a good correlation.

4) In the last part [hereafter, Paper (IV)], we are going to consider GRB time delays between γ-ray and radio Photons.

In-closing, we now give the synopsis of the present article. In Section 2, we make a brief review of the currently assumed sources that may be the cause of the observed time delays. In Section 3, we present the massive Photon model as it is widely understood. In Section 4, we present the rarefied cosmic plasma model that we believe explains the reason for the observed time delays in GRBs. Having presented our heads of argument as to what it is we hypothesize may be the reason for the time delays, in Section 5, we apply the proposed model to real data which has been procured from the literature, where this data is analysed and results are presented. Lastly, in Section 6 & Section 7, a general discussion is presented and conclusions are drawn, respectively.

There are about three major sources that may lead to the observed time delays, Δt, and these are, the effects of a Massive Photon, Plasma Effects and Intrinsic Processes associated with the Photon propagating in interstellar space. We will briefly discuss these effects below.

The common point of departure for most massive Photon theories is Maxwell-Proca Theory of Electrodynamics (MPED) [30]-[34]. In the MPED theory together with most of its variants, the energy-momentum dispersion of the Photon is given by Einstein’s [2] energy-momentum dispersion relation, namely:

where: E = h ν , p, and: m γ ≠ 0 , are the energy, momentum, and rest mass of the Photon respectively, while: h = 6.62606896 ( 33 ) × 10 − 34   J ⋅ s (CODATA 2018), is Planck’s constant and ν is the frequency of the Photon in question. Given that the group velocity, v g , of a wave is: v g = ∂ E / ∂ p . In the case where m γ ≡ 0 , we have that: v g = c 0 , while in the case: m γ ≠ 0 , we have that:

where: ν * = m γ c 2 / 2 ℏ , and, ℏ = h / 2 π = 1.054571628 ( 53 ) × 10 − 34   J ⋅ s , is Planck’s normalized constant.

It is easily seen from Equation (3) that the lower the frequency, the slower the Photon propagates in vacuo. For energetic events with short time scales such as GRBs, assuming Photons with different frequencies arriving at our telescopes are emitted simultaneously, the time delay of the low energy Photons relative to high energy ones thus can be used to calculate the rest mass of a Photon. In reality radiations of different bands arise at different times. For example, during a GRB explosion high energy Photons should be radiated earlier than X-ray to radio afterglows, and radio afterglows with higher frequencies emerge earlier than lower frequency ones. Therefore, by ignoring such intrinsic time delays, this method can be used to put an upper limit on the Photon rest mass.

If: D , is the distance between the Earth and the GRB, and v l and v h are the group velocities for the lower and higher frequency Photons, it follows that the time delay Δt, is such that:

thus, if: Δt, D , ν l , and ν h , are known, the mass m γ of the Photon can be computed (see e.g., Refs. [35][36][37][38]).

Now, the distance D is the Light-travel distance or the look-back time ( t = D / c 0 ). This distance is typically calculated (see e.g., [35][36][37][38]) by assuming a Friedman Universe—i.e., the expanding Universe within the framework of the standard Cosmological-Constant Cold-Dark-Matter model (ΛCDM-model); and that the redshift z is solely due to the expansion of the Universe, then, this distance D L is given by:

where: Ω m = 0.315 ± 0.007 , and: Ω Λ = 0.685 ± 0.007 , are the matter-density and the darkenergy-density parameters as currently measured, respectively; ℋ 0 = 67.40 ± 0.50   km ⋅ s − 1 ⋅ Mpc − 1 [39] is the present day Hubble parameter. Thus, in the present article we adopt the said values for: Ω m , Ω Λ , and: ℋ 0 , for the computation of the luminosity distances to the GRBs and their host galaxies. We assume a flat StandardΛCDM-Cosmology Model and for all our calculations of the luminosity distances to the different GRB’s and their host galaxies, we shall use Wright’s [40] cosmology distance calculator².

The one major problem and setback with this massive Photon theory [summed up in Equation (4)] is that—technically speaking—it cannot be falsified because any quadruple of values ( Δ t , D , ν l , ν h ) will yield a mass for the Photon since in this theory the mass of the Photon is assumed to be variable. While the Photon mass is assumed to be variable, there is no explicit method of knowing how this mass varies with frequency—this obviously is yet another weakness of the theory. If the explicit variation of the Photon mass with frequency [ m γ = m γ ( ν ) ] was known, this could be used as a critical yardstick to falsify the theory. As currently obtaining, the theory is correct all the way every time—all we must do is to accept with little or no qualms at all, the mass deduced from the theory—this surelyis a scientifically difficult thing to do. In addition to all these obvious weaknesses, the theory further assumes that the interstellar space through which these Photons propagate is a proper vacuum with a refractive index identically equal to unity. Beginning in the next section, we present our alternative view to this story. We strongly believe this view is new and is being presented for the first time.

Photons will propagate at the fundamental and sacrosanct Light speed: c 0 = 2.99792458 × 10 8   m ⋅ s − 1 , only in a perfect vacuum were the refractive index (n) is identically equal to unity ( n ≡ 1 ). On that pedestal of understanding, it is worthy asking if the Intergalactic Medium (IGM) is a perfect vacuum. Is the space between galaxies truly empty enough to constitute a perfect vacuum? A brutally frank and honest answer to this important question would be—“No, the IGM is certainly not a perfect vacuum—there are a number of reasons for this. Stars, pulsars, the Active Galactic Nuclei (AGN) etc are constantly pouring out and into the IGM charged particles.” However minute the quantities of matter being poured into the IGM; it makes a significant difference in making the IGM a non-vacuo medium.

Actually, the IGM is known to be a rarefied plasma (see e.g., [41][42]) consisting mostly of ionized hydrogen; i.e. a plasma consisting of statistically equal numbers of Electrons and Protons. Therefore, the refractive index of the IGM and cosmological space in general cannot be identically equal to unity because of this cosmological, galactic and astronomical rarefied plasma and the magnetic fields. In a such a medium, the speed of propagation of a Photon will certainly dependent on its wave-length as it does here in earth laboratories in the different mediums such as glass, water, salt solutions etc. Apart from the rarefied plasma, there exists in the IGM the Intergalactic Magnetic Fields (IGMFs) [43][44] and as-well cosmological Primordial Magnetic Fields (PMFs) [45][46].

Logically, it therefore makes sense to imagine or assume that the vastness of all the cosmological space of the observable Universe must be filled with a rarefied plasma. Actually, physicists working with GRBs and using the time delays to estimate stringent mass limits of the Photon do acknowledge (see e.g., [6][7][35][36][38]) the existence of plasma in the interstellar space. The only problem is that they (see e.g., [6][7][35][36][38]) argue that the Plasma Effect is negligible. The Plasma Effect they talk about is the effect on the propagation of Light due to the oscillations of the Electrons in the plasma.

Our theory is wholly drawn from Maxwell’s [1] equations of Electrodynamics and these equations are given by:

where: E , B , J and ρ e , are the electric and magnetic fields of the travelling Photon, the current density of the absorbed Electron and the charge density of the ISM respectively.

Now, taking the curl of Equation (6b) and (6d) and also making use of Equation (6a) and (6c) in our computation, we obtain:

where:

is the D’Alembert operator. These Equations (7a) & (7b) are the well known electromagnetic wave equations of motion for a Photon in a non-vacuo medium.

If—as is the case in Earth based laboratories—these electrical currents obey Ohm’s Law ( J = i σ E ) where σ is the conductance of this cosmic plasma, then, the wave equations for E and B will be given by (see e.g., Lorrain & Corson [47], p. 468):

where, in a perfect vacuum were σ ≡ 0 , we have that: □ E = 0 , and: □ B = 0 .

Now, assuming for, E , and, B , the electric and magnetic field wavefunctions: E = E 0 e − i k α x α , and, B = B 0 e − i k α x α , where E 0 and B 0 are constant vectors and k μ is the four wavenumber, then, these two wave Equations (9a) and (9b), yield the following dispersion relation:

where: ω * = 2 π ν * = μ c 2 / 4 , ω = 2 π ν , with ν being the frequency of the Photon and k its wavenumber. Given that the group velocity v g of a wave is given by: v g = ∂ ω / ∂ k , thus differentiating Equation (10) throughout with respect to k and rearranging, it follows that:

where: v p = ω / k , is the phase velocity. In a vacuum we have that: v g = v p = c 0 . This assumption (of: v g = v p ) can be extended to the scenario of a non-vacuum medium and so doing (i.e., maintaining this condition: v g ≠ v p , in the non-vacuum medium), one obtains:

From Equation (12), as before, if: D , is the distance between the Earth and the GRB, and v l and v h are the group velocities for the lower and higher frequency Photons, then—to first order approximation where from Equation (12) we have that: c 0 / v g ≃ 1 + ν * / ν , it follows that the time delay Δt, is such that:

Thus, if: Δt, D , ν l , and ν h , are known, the conductance of the interstellar space can be inferred from the value of ν * , since:

For a large enough data set, it is clear that if the laid down theory has any correspondence with physical and natural reality, then, a plot of: Δtvs ( ν l − 1 − ν h − 1 ) , for the same source (i.e., same D ) should—accordingly—yield a straight line graph with a slope equal to D ν * / c 0 . In next section, we shall present four such graphs for four GRB afterglows. We must say that—the result from these four graphs is promising, all that is needed is further corroboration from a convincingly large enough data set. If anything, what is required for a plot of: Δtvs ( ν l − 1 − ν h − 1 ) , are only three parameters, namely the time delay Δt, the frequencies ( ν l , ν h ) of the Photon pair.

In Section 5.1, we give our choice of the data that we use for the present work and in Section 5.2, we present the analysis and the results obtained thereof.

We have demonstrated that there exists a reasonable and strong inverse frequency [ Δ t ∝ ( ν l − 1 − ν h − 1 ) ] correlation between the observed time delays and the frequency of the Photons observed at the telescope. Of the four GRBs in our case study, not only does GRB 030329 give the best correlation, this interesting source has the most data points which make this result statistically significant. An closer inspection of Figure 1, will reveal that 14 GRB-events associated with GRB 030329 can be grouped into four distinct groups—i.e.: with the first group being GRB 030329a-f, the second being GRB 030329g-j, and the third being the lone event GRB 030329k and lastly the forth being GRB 030329i-o. The events: GRB 030329a-f, GRB 030329g-j and GRB 030329i-o, can be fitted neat straight lines that have nonzero y-intercepts. In Paper (II), we will consider these events independently, where upon, we shall see that these events (via their nonzero y-intercepts) suggest a non-simultaneous—albeit—well correlated emission between the ( ν l , ν h ) -signals.

In comparison to the Plasma and Photon Mass Effects, what is interesting is that, for the same GRB source, not only is the expected linear correlation between Δt and ( ν l − 1 − ν h − 1 ) independent of the distance to the source, but this relationship, unlike the the Plasma and Photon Mass Effect that have a ν − 2 variation, we here have a ν − 1 variation on the ( ν l , ν h ) -Photon frequencies. This is very important in that if this behaviour were to be confirmed for a statistically significant number of GRBs events, it would rule out the Plasma and Photon Mass Effect as possible candidates to the cause of these time delays. At any rate, this would be a significant step forward in our understanding of GRBs and the propagation of EM-waves in the cosmic ISM.

If the cosmic ISM were to be thought of as a conductive medium, then, from the values of the conductance here obtained [~(10⁻¹⁴ - 10⁻¹¹) Ω∙m], one can safely say that the cosmic ISM is a poor conductor of electricity; this of cause is expected. Metals have conductances whose magnitude is of the order ~10⁷ Ω∙m (see e.g., Refs. [60][61][62]). Clearly, from the above stated conductances [~(10⁻¹⁴ - 10⁻¹¹)Ω∙m] obtained herein, we see that the rarefied cosmic plasma must be a poor conductor of electricity—for, its conductance is at least twenty one orders of magnitude smaller compared to ordinary metals.

Now, with regard to the interaction mechanism between the Photon and the plasma in the present model, one will rightly ask: Since the Photon and the plasma are here interacting,what is different between this proposed interaction mechanism and the Plasma Effect? To that, we have the following to say. The Compton wavelength of Photon—or more so,its radius—is much smaller than the wavelength of radio waves. From an intuitive physical standpoint, it is possible to imagine an Electron being engulfed by the Photon in such a manner that the Electron can be pictured to be moving inside the E and B-fields of the Photon. Succinctly stated, the Electron is absorbed by the Photon in much the same manner as the Photon is absorbed by the Electron in such phenomenon as the Photo-electric effect [63], i.e.:

where γ − , is an electrically charged Photon that off-cause does not propagate at the speed of Light c in vacuo, but propagates at a lesser speed just as happens with ordinary material bodies. It must be said that, the Photon state “ γ − ” is not here envisaged as a permanent state of the Photon as it propagates in the ISM, but a relatively short-lived (transitory) state, just as the absorption of the Photon by the Electron can be a short-lived (transitory) state.

In the forward reaction (interaction): e − + γ → γ − , we have the Electron being absorbed by the moving Photon, in which process, the Photon acquires some inertia from the Electron, leading to the alteration of its speed in-accordance with the Equation (12), and, in the reverse reaction: γ − → e − + γ , the Electron is ejected from the Photon system and returns to the cosmic plasma medium from which it originated, in which event thereafter, it [Photon] travels as it would normally do—with the Plasma Effect perhaps coming into the picture. The “fact” that: Δ t ∝ ( ν l − 1 − ν h − 1 ) , strongly points in the direction of the here proposed “Electron absorption phenomenon” as being the dominate mechanism leading to the observed time delays, with the Plasma Effect having a negligible contribution to Δt. In-closing, allow us to say that: in the next article [i.e., Paper (III)] of our four part series, we shall consider the four GRBs that did not give a good correlation (namely, GRB 060218, GRB 000926, GRB 031203 and GRB 991208).

If what has been presented herein be considered reasonable or acceptable—one can on this basis—make the following tentative conclusion regarding these observed time delays:

1) The observed time delays may not (as is widely believed or assumed) originate from some supposed Lorentz violating mechanism, but, from a (seemingly not considered before) Plasma-like Effect to do with in-situ cosmic Electrons (and perhaps Protons as-well) in the supposed rarefied cosmic plasma interacting with the electric and magnetic components of the propagating Photon.

2) The suggested hypothetical interaction of the in-situ cosmic Electrons with the electric and magnetic components of the propagating Photon leads to a modification of the speed (group velocity) of the Photon through the ISM, wherein—it is seen that, the larger the frequency, the faster the Photon and vice-vesa, hence, the observed time delay.

We wish to acknowledge the financial support from the Education, Audio and Culture Executive Agency of the European Commission through the Pan-African Planetary and Space Science Network under funding agreement number 6242.24-PANAF-12020-1-BW-PANAF-MOBAF. Also, we could like to acknowledge the invaluable support from our work stations—The Copperbelt University (Republic of Zambia) and the National University of Science and Technology (Republic of Zimbabwe) for the support rendered in making this work possible.

No new data were generated in support of this research. As stated in the main text [i.e., Section 5.1], the data underlying this article were wholly derived from Zhang et al. [6].

In-order not to distrust the reader, we have placed here in the appendix all the data TablesA1-A4 for the four sources [GRB 030329, GRB 980425, GRB 000418 and GRB 021004] used to obtain the graphs [Figures 1-4, respectively] in the main text.

TableA1. Data for GRB 030329@(z = 0.169).

From this data—according to Figure 1, we obtain: D ν * c 0 = ( 9.20 ± 0.30 ) × 10 15 .

TableA2. Data for Data for GRB 980425@(z = 0.009).

From this data—according to Figure 2, we obtain: D ν * c 0 = ( 4.90 ± 0.30 ) × 10 15 .

TableA3. Data for Data for GRB 000418@(z = 1.119).

From this data—according to Figure 3, we obtain: D ν * c 0 = ( 7.60 ± 0.80 ) × 10 15 .

TableA4. Data for GRB 021004@(z = 2.330).

From this data—according to Figure 4, we obtain: D ν * c 0 = ( 1.50 ± 0.20 ) × 10 16 .

¹[3].

²http://www.astro.ucla.edu/%7Ewright/CosmoCalc.html: visited on this day: Sunday 5 Apr. 2019.

³The HETE-2 satellite [High Energy Transient Explorer (HETE)] is a small scientific satellite designed to detect and localize gamma-ray bursts. The HETE program is an international collaboration led by the Center for Space Research at the MassachussettsInstitute of Technology. The coordinates of GRBs detected by HETE are distributed to interested ground-based observers within seconds of burst detection, thereby allowing detailed observations of the initial phases of GRBs.

⁴On 26 October 2018, through an electronic vote conducted among all members of the International Astronomical Union (IAU), the resolution to recommend renaming the Hubble Law as the Hubble-LemaîtreLaw was accepted. This resolution was proposed in order to pay tribute to both—Georges Henri Joseph Édouard Lemaître (1894-1966), and, Edwin Powell Hubble (1889-1953), for their fundamental contributions to the development of the modern expanding cosmology model.