Planck Radiation Formula for Massive Photons (II)

Planck Radiation Formula for Massive Photons (II) G. G. Nyambuya1, 1 National

University of Science and Technology, Faculty of Applied Sciences — Department of Applied Physics, Fundamental Theoretical and Astrophysics Group, P. O. Box 939, Ascot, Bulawayo, Republic of Zimbabwe. E-mail: [email protected]

We present an improved derivation of the Planck Radiation Law for Massive Photons (PRLMP). Unlike the earlier derivation, the present derivation makes no approximations of any kind — it is exact. In addition to the T 2 term in the PRLMP, a new T 3 term (pole) emerges, i.e., for the Massive Photon StefanBoltzmann Law (MPSBL), we obtain: (εrad = ǫσ0 T 4 − ǫη3 σ0 T 3 + ǫη2 σ0 T 2 ). The coefficients of both the terms (T 3 & T 2 ), i.e. [(η3 ≥ 0) & (η2 ≥ 0)] contain in them — the mass of the Photon. The new MPSBL is applied to Carl` a [1]’s experimental data on four different Tungsten lamps. While the new version of the massive Photon theory is in total agreement with experimental data in that the MPSBL must be a fourth order polynomial, the resulting coefficients yield inconsistent masses for the Photon. These inconsistencies are most likely due to the fact that the emissivity of the three poles (T 4 , T 3 , T 2 ) of the new MPSBL have been assumed to be equal.

“Thus,

the Photons which constitute a ray of Light behave like intelligent human beings: out of all possible curves they always select the one which will take them most quickly to their goal.”

— Max K. E. L. Planck (1858-1947)

1

Introduction

In Ref. [2] [hereafter Paper (I)], we derived the Planck Radiation Law for Massive Photons (PRLMP) and from there-on, we proceeded naturally to derive an experimentally testable Massive Photon Stefan-Boltzmann Law (MPSBL). In this derivation of Paper (I), the hypothetical massive Photon is assumed to have a momentumdependent ‘rest mass’. In-order to have a momentumdependent rest mass of the Photon that is in conformity with observational evidence in that this hypothetical momentum-dependence be not sensitive as this would — contrary to evidence — lead to its detectability within the realm and regime of laboratory measurements, we employed an approximation in-order to meet this demand. However, in Ref. [3] [hereafter Paper (II)], the resulting MPSBL of Paper (I) was tested using Carlà [1]’s experimentally derived data where it was seen that the mass of the Photon is generally non-zero. The problem with this non-zero mass Photon of Paper (II) is that this mass appears to not only be variable, but also imaginary as-well. In-order to overcome this setback and come up with something more believable, we noted upon a meticulous and close inspection, that, the approximation that we made in-order for the momentum-

dependent ‘rest mass’ to be not sensitive within the realm and regime of laboratory measurements, this assumption is not necessary. Therefore, we drop this assumption here and thereafter proceed to re-derive a new PRLMP and hence as new MPSBL. 2

New Derivation of the PRLMP

It has been demonstrated in Refs. [4, 5], that starting from the usual Einstein energy-momentum relation: 2 4 E 2 = p2 c2 + mE c ,

(1)

where (E, p) are the particle’s energy and particle’s momentum respectively, c the speed of Light in vacuo, and mE is the Einstein mass1 of this same particle, and hypothetically assuming a momentum dependent Einstein mass i.e. mE = mE (p), and further treating particles as waves, the meaning of which is that the group velocity of these waves is given by: vg =

∂E , ∂p

(2)

the energy of such particles is such that: 1 We have written the Einstein [6] rest mass using the symbol mE and not the traditional symbol m0 . The reason for this is deliberate. Einstein’s m0 represents the mass of the particle when it is ‘stationery’ i.e. when p = 0. In our model, this is not the case, so, by writing m0 as mE , we would like to make this distinction clear from here-on. We shall call mE the Einstein mass and not the rest mass. Latter on, we shall denote the rest mass using the symbol m∗ .

G. G. Nyambuya. Planck Radiation Formula for Massive Photons (II)

1

2 pc2 c4 ∂ 2 mE E= + . vg 2vg ∂p

(3)

dN =

8πV p2 dp 8πV (pc)(pdp) 2V d3 p = = , 3 h h3 h3 c

(10)

This relation Eq. (3) is obtained by differentiating Eq. (1) throughout with respect to p. Furthermore, differen- where h is Planck’s constant. Substituting ‘pdp’ as given tiating Eq. (3) throughout with respect to p, one obtains: in Eq. (9) and ‘p’ in-terms of E as given in Eq. (7) into Eq. (10), we will have: 2 c4 ∂ 2 mE 2 2 . (4) vg = c + 2 ∂p2 m c 8πV hν ∗ 2 dN = dhν . dhν − hν − m c ∗ In the case of electromagnetic waves, we have vg = c, h3 c c2 c thus substituting vg = c, into Eq. (4), one obtains (11) 2 ∂ 2 mE /∂p2 = 0, and as has been argued in Ref. [5], this Unpacking the terms in the square-brackets in Eq. (11), implies that: we will have: p 2 2 2 , (5) mE = m∗2 1 − m c 8πV h ν ∗ p∗ dN = hνdhν dhν − 2 h3 c c2 c . where in this model, m∗ is the rest mass of the particle 2 2 + m∗ c dhν — this mass is equal to the Einstein mass only when (12) p = 0, i.e., when the particle is at ‘rest’, hence our redefinition of the rest mass. The reader ought to make a We know that the actual number of occupied states dn clear distinction between mE and m∗ . is such that [dn = fBE (ν, T )dN ], where: Now, if the Einstein mass is given as above and we 1 needed it to not be sensitive in the current regime of fBE (ν, T ) = hν/k T , (13) B e −1 laboratory measurements, it would mean we would have to make p∗ large compared to p, i.e. p ≪ |p∗ |. We find is the Bose-Einstein probability function — which — for that if we choose p∗ such that p∗ = m∗ c, the assumption a temperature T , it gives the probability of occupation p ≪ |p∗ |, will not be required any-more as this would of a quantum state whose energy is E = hν. From the lead to the energy E of the Photon now being given by: foregoing, it follows that: h2 ν 2 dhν 8πV E = pc + m∗ c2 ⇒ pc = E − m∗ c2 . (6) dn = 3 3 h c ehν/kB T − 1 hence: m c E2 16πm∗ V hνdhν ∗ 2 2 p = 2 −2 E + m∗ c . (7) − , (14) 3 hν/k BT − 1 c c h c e With this assumption that p∗ = m∗ c, in-order to meet the demand that the momentum dependence of the Eindhν 8πm∗2 cV + stein mass be not sensitive in the regime of laboratory h3 ehν/kB T − 1 measurement, all we have to have is that m∗ be extremely hence, the energy density [u(ν, T )dν = hνdn/V ] is given small. Now, from Eq. (6), substituting E = hν where ν is by: the frequency of the massive Photon, it follows that: u(ν, T )dhν h3 ν 3 dhν 8π = 2 2 h ν m∗ c h h3 c3 ehν/kB T − 1 p2 = 2 − 2 hν + m∗2 c2 , (8) c c hence: h2 ν 2 dhν 16πm∗ . (15) − 3 hν/k BT − 1 h c e hν m∗ c pdp = dhν. (9) dhν − c2 c hνdhν 8πm∗2 c + Going back to the basics of our derivation of a PRLMP, 3 hν/k BT − 1 h e we know that the number of quantum states dN in the momentum volume space d3 p and physical volume space Now, setting x = hν/kB T : this implies dhν = kB T dx, thus substituting this into the above, we will have: V , is given by: 2

2

2 New Derivation of the PRLMP

−

+

We have to take into account the background radiation by including the Newtonian cooling term [ǫη1 A (T − Tbg )] into the above equation by adding this term to it, i.e.:

4 4 8πkB T 3 h c3

x3 dx ex − 1

3 3 16πm∗ kB T h3 c

x2 dx . ex − 1

2 2 8πm∗2 ckB T 3 h

ex

u(ν, T )dhν = h

3 4 − ǫη3 σ0 T 3 − Tbg εnet = ǫσ0 T 4 − Tbg

(16)

. 2 + ǫη1 (T − Tbg ) +ǫη2 σ0 T 2 − Tbg

xdx −1

(24)

Hence, from this equation, it follows that:

Further, the total energy density ε is such that: c ε= 4

∞

Z

0

ckB T u(ν, T )dν = 4h

Z

3 4 − ǫη3 σ0 A T 3 − Tbg Pnet = ǫσ0 A T 4 − Tbg u(x)dx,

(17)

0

This equation can be written in a much simpler form as:

hence, from the foregoing, it follows that: ε=

−

+

4 4 2πkB T 3 h c2

3 3 4πm∗ kB T 3 h

2 2 2πm∗2 c2 kB T 3 h

Given that: Z

∞

0

R

x3 dx ex − 1

∞

0

R

∞

0

Z

∞

0

R

∞

0

R

∞

x3 dx ex − 1

Pnet = aT 4 − bT 3 + cT 2 + dT − e,

x2 dx . ex − 1

(18)

xdx −1

a b c d e

3

=

2!ζ(3) ≃

xdx ex − 1

=

π2 , 6

= = = = =

ǫσ0 A, ǫη3 σ0 A = ǫη2 σ0 A = ǫη1 A = 4 3 2 aTbg − bTbg + cTbg + dTbg .

Formula for Calculating the Photon Mass

We here present the formula to calculate the mass of the (19) Photon. From Eq. (27), we have that (c/b = η2 /η3 ), and from Eq. (21) & (22), hence it follows that:

12 , 5

5π 3 η2 = η3 10368

it follows that:

(20) m∗ (η2 , η3 ) =

where: 1 σ0

1 η2 = σ0

m∗ c2 kB

,

(28)

hence:

εrad = σ0 T 4 − η3 σ0 T 3 + η2 σ0 T 2 ,

η3 =

aη3 , aη2 , aη1 /σ0 ,

(27) Our derivation ends here. We will — in the next section — discuss this very important result, i.e., Eq. (26).

π4 , 15

x2 dx ex − 1

(26)

where:

ex

0

=

. (25) 2 + ǫη1 A (T − Tbg ) +ǫη2 σ0 A T 2 − Tbg

∞

72 π

m∗ c2 kB

5π = 2

m∗ c2 kB

2

3 48πkB 5h3

m∗ =

2 π 3 c2 kB 3h3

m∗2

5π 3 c kB . 10368 a c2

(29)

The values of a and c are known from the curving pa(21) rameters, hence the mass of the Photon can be deduced. This was shall do in the next section.

,

.

(22)

m∗ (η2 ) =

m∗ (η3 ) =

r

πb 72a

kB . c2

(30)

2c kB . 5πa c2

(31)

Correcting for emissivity, we will have: εrad = ǫσ0 T 4 − ǫη3 σ0 T 3 + ǫη2 σ0 T 2 .

(23)


3

Fig. 1: Power vs Temperature Graphs for the Four Lamps: In the Figure above are the Power vs Temperature graphs for the four lamps and superimposed on these graphs are the fitting curves to the data points. The resulting fitting parameters are presented in Table (1).

4

Comparison with Carl` a (2013)’s Data

there are moments of despair and in our own endeavour We here have derived a new Stephan-Boltzmann Law for to building on our earlier ideas [4, 5] leading to a consismassive Photons where this new relationship is a com- tent massive Photon theory, this moment where we obplete fourth order polynomial in-terms of T . What does tained different values of m∗ (η2 , η3 ), m∗ (η2 ) and m∗ (η3 ), experimental data have to say about this? This is some- proved to be our most desperate moment and it is at this thing that we need to check before we intertain this new moment that we where visibly reminded of — and comresult. The way to go about it is to check if a plot of P forted by — Albert Einstein (1879-1955)’s own words: vs T yields a polynomial fit of the fourth order. For this “As for the search for truth, I know from my we shall use Carlà [1]’s data for the four lamps. own painful searching, with its many blind alleys, As one can verify for themselves from Figure (1), the how hard it is to take a reliable step, be it ever so experimental data of Carlà [1] agrees 100% that the P small, towards the understanding of that which is vs T curve is a fourth order polynomial — the goodtruly significant . . . the years of searching in the ness to fit correlation coefficient (R2 ∼ 100%). Table (1) dark for a truth that one feels but cannot express, presents the fitting parameters, while Table (2) presents the intense desire and the alternations of confithe disappointing Photon mass as deduced from the said dence and misgiving until one breaks through to fitting parameters of Table (1). In-accordance with Eq. clarity and understanding, are known only to him (27), column (2) of Table (2) is obtained by dividing a who has experienced them himself.” by the SB-constant σ0 i.e., ǫA = a/σ0 , while column (3) and (4) are obtained as follows: η3 = b/a and η2 = c/a. We felt that the attainment here of a fourth order polyThe masses m∗ (η2 , η3 ), m∗ (η2 ) and m∗ (η3 ) are then com- nomial MPSBL (which is in total agreement with the curve obtained from the data obtained from experiment) puted from Eq. (29), (30) and (31), respectively. If our massive Photon theory is to be believed, logic was a subtle hint and a tacit ray of Light pointing to dictates that m∗ (η2 , η3 ) = m∗ (η2 ) = m∗ (η3 ). The result ‘fact ’ that somehow, we where on the right track, all we in Table (2) goes against all such logic. That is to say, needed was to take one more ‘leap of faith’. It is at this point that we decided to find subtle against all expectations, these masses m∗ (η2 , η3 ), m∗ (η2 ) and m∗ (η3 ) are different. If there is no way to remedy the and esoteric loopholes in our derivation — we pondered situation, then, one will have no choice but to watch the deeply on it (derivation). No sooner than expected, it ocentire edifice (that we have been building thus far) come curred to us that the problem lie in the way in which the down crashing. We would sadly have to throw this theory emissivity has been introduced in this new MPSBL. The out of the window and move-on with our head nodded emissivity of the three poles of the suggested MPSBL i.e. facing the Earth in uttermost humility and humiliation. T 4 , T 3 and T 2 have been assumed to be equal and this All our once promising endeavours go up in smoke just may be the source of the problem. We will demonstrate in the sequential instalment (Ref. [7]) that this may very like that. As happens in all endeavours in building a theory, well be the case and therein (Ref. [7]), we not only ob4

4 Comparison with Carl` a (2013)’s Data

Table 1: Curve Fitting Parameters for the Four Lamps

Lamp

2.0V 3.3V 5.2V 6.2V

a (10−13 Js−1 K−4 )

b (10−10 Js−1 K−3 )

c (10−7 Js−1 K−2 )

12.30 ± 0.30 4.00 ± 5.00 3.85 ± 0.04 2.01 ± 0.03

−14.00 ± 2.00 −6.00 ± 5.00 −4.00 ± 0.20 −2.30 ± 0.10

16.00 ± 3.00 9.00 ± 10.00 6.20 ± 0.30 4.00 ± 0.20

d (10−4 Js−1 K−1 )

e (0.1Js−1 )

7.00 ± 3.00 5.00 ± 7.00 4.50 ± 0.20 1.90 ± 0.10

−3.10 ± 0.80 −3.00 ± 3.00 −1.96 ± 0.04 −0.97 ± 0.03

Table 2: Photon Mass

Lamp

2.0V 3.3V 5.2V 6.2V

η2

η3

(10−5 m2 )

(103 K)

(106 K2 )

2.18 ± 0.06 0.80 ± 0.80 0.69 ± 0.01 0.35 ± 0.01

1.20 ± 0.10 2.00 ± 3.00 1.04 ± 0.04 1.15 ± 0.05

1.09 ± 0.05

ǫA

m∗ (η2 )

m∗ (η3 )

m∗ (η2 , η3 )

(10−39 kg)

(10−38 kg)

(10−36 kg)

1.30 ± 0.30 2.00 ± 5.00 1.61 ± 0.09 2.00 ± 0.10

8.00 ± 1.00 10.00 ± 20.00 7.00 ± 0.40 7.70 ± 0.60

6.30 ± 0.80 8.00 ± 9.00 7.00 ± 0.20 7.70 ± 0.30

3.00 ± 0.70 4.00 ± 10.00 3.70 ± 0.20 4.60 ± 0.30

1.70 ± 0.10

7.30 ± 0.30

7.20 ± 0.20

3.90 ± 0.20

tain a consistent and coherent Photon mass, but values These three different poles (T 4 , T 3 , T 2 ) have co-related, of the emissivity and effective surface areas of Carlà [1] albeit, different emissivities. It is these different emissivfour lamps. ities that have led to the different values of the mass of the Photon leading to a prima facie case of a logically incorrect and incoherent theory. 5 General Discussion We here have derived a new SBL for massive Photons. Our present new derivation is correct on one account, that is, on the account that the massive Photon theory is in total agreement with experimental data in that the resulting MPSBL must be a fourth order polynomial with the poles (T 4 , T 3 , T 2 ). On the account of the resulting Photon mass emerging from the coefficients of these poles (T 4 , T 3 , T 2 ), the theory — as presently constituted — is wrong. While this may be the case that the theory is wrong, there — however — is hope that this shortcoming can be overcome. As will be demonstrate in the sequential instalment (Ref. [7]), these inconsistencies are most likely due to the fact that the emissivity of the three poles (T 4 , T 3 , T 2 ) of the new MPSBL have been assumed to be equal. Thus, no tangible conclusion can be drawn from the present reading. All we can say is that this reading is a pathway to our final destination, which is the sequel reading Ref. [7], where we shall present arguments to the effect that the seemingly most natural, apparently most normal and most straight forward way we have introduced the emissivity in the new MPSBL is at fault.

References 1. M. Carl` a. Stefan-Boltzmann Law for the Tungsten Filament of a Light Bulb: Revisiting the Experiment. American Journal of Physics, 81(7):512–517, Jun. 2013. 2. G. G. Nyambuya. Planck Radiation Formula for Massive Photons (I). Prespacetime Journal, 8(5):663–674, Jun. 2017. 3. G. G. Nyambuya. Photon Mass Determination from a Typical Stefan-Boltzmann Freshman-Type Experiment. Progress in Physics, ***(***):***–***, Jul. 2018. doi: 10.13140/RG.2.2.29558.70721. (Submitted). 4. G. G. Nyambuya. Gauge Invariant Massive Long Range and Long Lived Photons. Journal of Modern Physics, 5(17):1902– 1909, Nov. 2014. 5. G. G. Nyambuya. Light Speed Barrier as a Cosmic Curtain Separating the Visible and Invisible Worlds. Prespacetime Journal, 8(1):115–125, 11 Feb. 2017. 6. A. Einstein. Zur Elektrodynamik bewegter K¨ orper. Annalen der Physik, 322(10):891–921, 1905. 7. G. G. Nyambuya. Photon Mass from an Emissivity Corrected Planck Radiation Law for Massive Photons. Progress in Physics, ***(***):***, Jul. 2018.


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