Vol.3, No.4, 334-338 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.34044
Using of the generalized special relativity (GSR) in
estimating the neutrino masses to explain the
conversion of electron neutrinos
Mahmoud Hamid Mahmoud Hilo1,2
1Department of Physics, Faculty of Education, Al-Zaiem Al-Azhari University, Om Durman, Sudan;
2Department of Physics, Faculty of Science and Arts, Qassim University, Qassim, Kingdom of Saudi Arabia.
Received 4 March 2011; revised 22 March 2011; accepted 26 March 2011.
ABSTRACT
In this work the Generalized Special Relativity
(GSR) is utilized to estimate masses of some
elementary particles such as, neutrinos. These
results are found to be in conformity with ex-
perimental and theoretical data. The results ob-
tained may explain some physical phenomena,
such as, conversion of neutrinos from type to
type when solar neutrino reaches the Earth.
Keywords: Generalized; Neutrino Masses;
Conversion; Phenomena
1. INTRODUCTION
The concept of mass plays an important role in phys-
ics. The mass is very important property of any sub-
stance and element. The mass of any element or elemen-
tary particle is utilized to characterize this element or
particle. The masses of the elementary particles emitted
from the stars can be utilized to understand the nuclear
reactions and the mass conversions processes [1,2]. They
can also utilized to study the gravitational waves [3] as
well as supernova [4,5].
There are many problems associated with mass, espe-
cially in the world of the microscopic particles such as
photons, electrons, neutrinos and protons. These prob-
lems appeared in the middle of the 20th century. Many
physicists tried to solve the problems of mass [6,7]. The
most famous one is the neutrino mass problem. Neutrino
mass problem arises from the fact that the neutrino mass
is strongly dependent on the elementary particle associ-
ated with it. It was discovered that electron, muon, and
tau neutrino masses are different from each other [8,9].
This discovery is confirmed theoretically and experi-
mentally by using very sensitive detectors [10].
Neutrinos have electric charge, interact very rarely
with matter, and according to the text book version of
the Standard Model (SM) of particle physics – are mass
less. For every hundred billion solar neutrinos that pass
through the Earth, only about one interacts at all with
stuff of which Earth is made. There are three known
types of neutrinos. Nuclear fusion in the Sun produces
only neutrinos that are associated with electrons, the
so-called electron neutrinos (e

). The two other types of
neutrinos, muon neutrinos (


) and tau neutrinos (


),
are produced, for example, in laboratory accelerators or
in exploding stars, together with heavier versions of the
electron, the particles muon (μ) and tau (τ) [11].
Evidence obtained indicated that something must
happen to the neutrinos on their way to detectors on
Earth from the interior of the Sun. In 1990, Hans Bethe
and John N. Bahcall pointed that new neutrino physics,
beyond what was contained in the Standard Model parti-
cle physics textbook, was required to reconcile the re-
sults of the Davis Chlorine experiments and the Japanese
-American water experiments. This lead directly to the
relative sensitivity of the Chlorine and water experi-
ments to neutrino number and energy. The newer Solar
neutrino experiments in Italy and in Russia increased the
difficulty of explaining the neutrino data without in-
volving new physics [12]. On June 2001 the mystery of
Solar neutrino had solved by a collaboration of Canadian,
American, and British scientists. They reported the first
Solar neutrino results obtained with a detector of 1000
tons of heavy water (D2O). The new detector was able to
study in different way the same higher-energy Solar neu-
trinos that had been investigated previously in Japan
with the Kamiokande and Super – Kamiokande ordinary
– water detectors. The Canadian detector is called SNO
for Solar neutrino observatory [13].
The SNO collaboration made unique new measure-
ments in which the total number of high energy neutri-
nos of all types was observed in the heavy water detector.
These results from the SNO measurements alone show
that most of the neutrinos produced in the interior of the
Sun, all of which are electron neutrinos by the time they

M. H. M. Hilo / Natural Science 3 (2011) 334-338 335
reach the Earth. The solution of the mystery of missing
Solar neutrinos is that neutrinos are not, in fact, missing.
The previously uncounted neutrinos are changed from
electron neutrinos (e

) into muon and tau neutrinos that
are more difficult to detect [14].
The Standard Model of particle physics assumes that
neutrinos are mass less. In order for neutrino oscillations
to occur, some neutrinos must have masses. Therefore,
the Standard Model of particle physics must be revised.
The simplest model that fits all the neutrino data implies
that the mass of the electron neutrino is about 100 mil-
lion time smaller than the mass of the electron.
31
39
8
9.11 109.11 10
10
e
m



 (1)
But, the available data are not yet sufficiently defini-
tive to rule out all but one possible solution. When we
finally have a unique solution – as we will later, the val-
ues of the different neutrino masses may be clues that
lead to understanding physics beyond the Standard
Model of particle physics [15]. Neutrino mass given in
many previous studies are as follow, all neutrino masses
are given in the unit of eV. Experimental data shows that
the neutrino mass is given as follows (see Table 1):
The best neutrino mass limits from direct measure-
ments come from the tritium endpoint experiments
Mainz and Troitsk [16,17], both of which place m

<
2.2 eV. Measurements of the cosmic microwave back-
ground, coupled with cosmological models, have led to
somewhat better (but model-dependent) constraints of
m

< 1 eV [18]. The next generation of tritium endpoint
measurement is now being pursued by the KATRIN ex-
periment [19]. They expect to push the limit on the neu-
trino mass as low as m

< 0.2 eV. An independent
avenue of research is neutrinoless double-decay, which
could test the Majorana nature of the neutrino and possi-
bly determine its mass [20].
Also theoretical works presented different values of
the neutrino mass as follows:
The result e
m

< 4.5 eV obtained by. I. Hassan [21]
and the correspondent muon and tau neutrino masses
presented are:
63.4 eV, 225 eVmm
 
 (3)
Then Reference [17] presents e
m

< 10 eV and the cor-
respondent muon and tau neutrino masses presented are:
31.88 eV, 110 eVmm
 
 (4)
2. AIMS OF THE WORK
The aims of this work is to use the Generalized Spe-
cial Relativity (GSR) to estimate the neutrino masses so as
Table 1. Experimental data of neutrinos mass values obtained
in works [16-20].
work Electron neutrino Mass (eV)
[16,17] e
m

<2.2 eV
[18] e
m

<1 eV
[19] e
m

<0.2 eV
[20] e
m

<5.6 eV
to explain the problems associated with their masses, and
to explain the conversion of neutrinos from type to type.
Then to compare the result obtained using the (GSR)
theory with other theoretical and experimental works.
3. GENERALIZED SPECIAL RELATIVITY
(GSR) THEORY
The Generalized Special Relativity theory is a new
form of the special relativity theory that adopts the
gravitational potential, and it gives the formula of rela-
tive mass to be as follows [23]:
00 0
2
00 2
gm
m
v
gc


(5)
where 00 2
2
1gc

 , and

denotes the gravitational
potential, or the field in which the mass is measured .
The derivation of the mass Eq.5 using the generalized
special relativity (GSR) can be found as follows:
In the special relativity (SR), the time, length, and
mass can be obtained in any moving frame by either
multiplying or dividing their values in the rest frame by
a factor

.
2
2
1v
c
  (6)
where v is the velocity of the particle, and c is the speed
of light.
It is convenient to re-express

in terms of the
proper time, associated with the impact of gravity on the
previous physical quantities, (time, length, and mass)
[23].
22
dddcgxx



  (7)
where
g


is the metric tensor, and,

and

de-
notes the contra variant (covariant) vectors.
Which is a common language to both special relativity
SR, and general relativity (GR). We know that in special
relativity (SR) Eq.7 reduces to: [23].
22 220
dddd,
ii
cctxxxct 

 (8)

M. H. M. Hilo / Natural Science 3 (2011) 334-338 336
where i denotes the particle position (covariant) vector
according to Lorentz covariance.
2
22
d1dd
1.. 1
ddd
ii
xx v
ttt
cc


  (9)
Thus we can easily generalize

to include the effect
of gravitation by using Eq.7 and by adopting the weak
field approximation where [23].
1122 33002
2
1, 1 gggg c

 (10)
2
00 00
22
d1dd
..
ddd
ii
x
xv
gg
ttt
cc


   (11)
When the effect of motion only is considered, the ex-
pression of time in the special relativity (SR) is found to
be [23].
0
2
2
d
d
1
t
t
v
c


(12)
where the subscript 0 stands for the quantity measured in
the rest frame. While if gravity only affect time, its ex-
pression is given by [23].
0
00
d
dt t
g
 (13)
In view of Eqs.12, 13 and 11 the expression
0
d
dt t

 (14)
can be generalized to recognize the effect of motion as
well as gravity on time, to get
0
2
00 2
d
dt t
v
gc


(15)
The same result can be obtained for the volume where
the effect of motion and gravity respectively gives [23].
2
02
1v VV c  (16)
0000
VgV gV (17)
The generalization can be done by utilizing Eq.11 to
find that
2
000 0
2
v
VV gV
c
   (18)
To generalize the concept of mass to include the effect
of gravitation we use the expression for the Hamiltonian
in general relativity, i.e. [23].
2
0
200
0000 0
22
00
00 00
22
0
d
d
x
HcgTg
cmc
gg
V





 


(19)
where H is Hamiltonian,

is the density, and 00
T is
energy tensor.
Using Eqs.18,19, yields:
2
2
200 0
g
mc mc
cVV   (20)
Therefore
00 0
2
00 2
gm
m
v
gc


(21)
Which is the expression of mass in the presence of
gravitational potential and it named the generalized spe-
cial relativity (GSR) theory.
In view of Eq.5, and when we substitute the value of
00
g
, then the relative mass according to (GSR) is found
to be
02
2
22
2
1
2v
1
mc
m
cc








(22)
When the field is weak in the sense that
2
21
c

 (23)
And when the speed v is very low such that
2
21
v
c
 (24)
Eq.22 reduces to:
02
02
2
2
12
1
2
1
mc
mmm
c
c





 





(25)
Using the identity

11.for 1
n
xnxx  one can
also gets:
02
1mm c





(26)
And, when the field is so strong such that
2
22
21 and 1 v
cc

 (27)
Then Eq.22 reduces to

M. H. M. Hilo / Natural Science 3 (2011) 334-338 337
02
2
mm c

 (28)
Some gravitational models [21] propose the existence
of short range gravitational field. The expression of the
field potential is assumed to be:
1
0ecr
c
r


 (29)
For very small distance, i.e. when r → 0, Eq.29 re-
duces to:
0s
cGm
rr
  (30)
where Gs stands for the strong gravitational constant, to
find Gs one can assume that the strong gravity field is
the strong nuclear force itself. In this case we can use the
nuclear proton potential φp to find Gs. Where:
s
p
p
Gm
r
  (31)
4. RESULTS AND DISCUSSION
Let us recall Eq.2 8 and estimat the results of neutrino
masses. First of all one needs to find the strong gravity
constant Gs by using Eq.31 and substituting by the fol-
lowing values:
27 15
13
1.67 10kg,1.32 10m,
1.6710Nm kg
pp
p
mr


 

to get:
Gs = 1.23 × 1025 Nm2/kg2.
but since the electron radius and mass are given by:
33 31
10m, 9.1110kg,
ee
rm


Then from Eq.31 we get:
27
11.0710Nm kg
e
 
Using Eq.28 the electron neutrino zero mass can be
adjusted to obey the relation.
37
6.24 eV1.1110kg
e
m


 (32)
That means the mass of the electron neutrino (e m

) is
about million times smaller than the mass of the electron.
To calculate the mass of the muon neutrino (m


),
while the muon mass m

compared to the electron mass
is given by:
105.7 207
0.511 e
mm


So that the mass of muon neutrino is given by:
2
36
2207
89.78 eV1.5910kg
ee
mm m
c
m

 





(33)
To calculate the mass of the tau neutrino (m


), while
the tau mass m

compared to the electron mass is given
by:
1784 3491
0.511 e
mm


So that the mass of muon neutrino is given by:
2
34
23491
368.7 eV6.5510kg
ee
mm m
c
m

 





(34)
5. CONCLUSIONS
The mass values found by Eqs .3 2-34 using the (GSR)
theory, for the three kinds of neutrinos compared to the
values found by the different works as in Eqs.1-3, shows
that the change in the neutrino mass is attributed in this
model as resulted from the effect of the electron, muon,
and tau strong gravitational field on the neutrino mass.
The neutrino masses obtained due to the effect of e,

and

gravitational field are given in Table 2. The ar-
rangements of these values are in agreement with that
obtained in different experimental works in the ranges
that given in Reference [17-20], and theoretical works
[21]. A direct comparison between the values obtained
using the generalized special relativity (GSR) model and
the experimental and theoretical values shows that they
are in conformity with each other. Experimental data
found in different works agree that the neutrino mass
estimated is the mass of electron neutrinos, which was
able to detect in many detectors such as SNO collabora-
tions [14]. The mass value of correspondent neutrinos
(muon and tau), was calculated as a value that depend on
the relation between the masses of the particles (electron,
muon, and tau) themselves, as shown in the Results sec-
tion. So the values found in this work using the genera-
Table 2. Neutrinos mass values obtained using the GSR model.
particle Mass symbolMass (Mev) Mass (kg)
Electron e
m 0.511 31 9.1110  
Electron
neutrino e
m

6.24 eV 37
1.1110  
Muon m

207
e
m 28 1.8910  
Muon
neutrino m

89.78 eV 34 1.5910  
Tau m

3491
e
m 27
4.5710 
Tau neutrinom


368.7 eV 34 6.5510  

M. H. M. Hilo / Natural Science 3 (2011) 334-338 338
lized special relativity (GSR) model, affirm that the
missing neutrinos are not, in fact, missing, but, actually
are not able to detect. And that explains the conversion
of a part of the electron neutrinos into muon and tau
neutrinos, by the time that Solar neutrinos reach the
earth, which explain why the number of detected neutri-
nos is less than that predicted by theoretical model of the
sun and by textbook description of neutrinos.
REFERENCES
[1] Nadyozhin D.K. (1978) The neutrino radiation for a hot
neutron star formation and the envelope outburst problem.
Astrophysics and Space Science, 53, 131-153.
doi:10.1007/BF00645909
[2] Mikheev S.P. and Smirnov A.Y. (1985) Resonance en-
hancement of oscillations in matter and solar neutrino
spectroscopy. Soviet Journal Nuclear Physics, 42, 913-
917.
[3] Kotake K., Sato K. and Takahashi K. (2006) Explosion
mechanism, neutrino burst, and gravitational wave in
core collapse supernovae. Reports Progress in Physics,
69, 971-1144.
doi:10.1088/0034-4885/69/4/R03
[4] Janka H.T. et al. (2007) Supernova explosions and the
birth of neutron stars. McGill University, Montreal.
DOI: 10.1063/1.2900257.
[5] Colgate S.A. and White R.H. (1966) The hydrodynamic
behavior of supernovae explosions, astrophysics. Astro-
physical Journal, 143, 626-681.
[6] Jegerlehner B., et al. (1996) Neutrino oscillations and the
supernova 1987A signal. Physical Review D, 54, 1194-
1203. doi:10.1103/PhysRevD.54.1194
[7] Lunardini C. and Smirnov A.Y (2001) Neutrinos from
SN1987A, Earth matter effects and the LMA solution of
the solar neutrino problem, Physical Review D, 63,
073009. doi:10.1103/PhysRevD.63.073009
[8] Lunardini C. and Smirnov A.Y. (2004) Neutrinos from
SN1987A: Flavor conversion and interpretation of results,
Astroparticle Physics, 21, 703-720.
doi:10.1016/j.astropartphys.2004.05.005
[9] Aliu E. et al. (2005) Evidence for muon neutrino oscilla-
tion in an accelerator-based experiment. Physical Review
Letter, 94, 081802.
doi:10.1103/PhysRevLett.94.081802
[10] Esteban-Pretel A. et al. (2008) Mu-tau neutrino refraction
and collective three-flavor transformations in supernovae.
Physical Review D, 77, 065024.
doi:10.1103/PhysRevD.77.065024
[11] Abdurashitov J.N. et al. (2002) Measurements of the
solar neutrino capture rate by the Russian-American gal-
lium solar neutrino experiments during one half of the
22-year cycle of solar activity. Journal of Experimental
and Theoretical Physics, 95, 181-193.
doi:10.1134/1.1506424
[12] Drell S. (2003) Personal letter to John Bahcall, January
29. Soviet Journal Nuclear Physics, 42, 913-917.
[13] Bahcall J.N. et al. (2001) Solar models: Current epoch
and time dependences, neutrinos, and helioseismological
properties. Astrophysical Journal, 555, 990-1012.
doi:10.1086/321493
[14] Eguchi K. et al. (2003) First results from kamland: Evi-
dence for reactor antineutrino disappearance. Physical
Review Letter, 90, 021802.
doi:10.1103/PhysRevLett.90.021802
[15] Bondi, H. (1957) Negative mass in general relativity.
Reviews of Modern Physics, 29, 423-428.
doi:10.1103/RevModPhys.29.423
[16] Bonn J. et al. (2002) Results from the Mainz neutrino
mass experiment. Progress in Particle and Nuclear Phys-
ics, 48 , 133-139.
doi:10.1016/S0146-6410(02)00119-9
[17] Lobashev V.M. et al. (2002) Study of the tritium
beta-spectrum in experiment “Troitsk ν-mass”. Progress
in Particle and Nuclear Physics, 48, 133-139.
doi:10.1016/S0146-6410(02)00118-7
[18] Komatsu E. et al. (2009) Five-year wilkinson microwave
anisotropy probe (WMAP) observations: cosmological
interpretation. The Astrophysical Journal Supplement Se-
ries, 180, 330-376.
[19] Bornschein L. (2003) Collaboration KATRIN-direct me-
asureement of neutrino masses in the SUB-EV region.
Universität Karlsruhe, Karlsruhe.
[20] Elliott S. and Engel J. (2004) Double beta decay. Journal
of Physics G: Nuclear and Particle Physics, 30, R183
–R215. doi:10.1088/0954-3899/30/9/R01
[21] Hassan I. (1998). Neutrino masses. M.Sc. thesis, Sudan
University of Science and Technology, Khartoum.
[22] Braginsky V.L., et al. (1957) Laboratory experiments to
test relativistic gravity. Physical Review D, 15, 2047-
2068. S. Chu. doi:10.1103/PhysRevD.15.2047
[23] Hilo M.H.M. et al. (2011) Using of the generalized spe-
cial relativity in estimating the proton (nucleon) mass to
explain the mass defect. Natural Science, 3, 141-144.
doi:10.4236/ns.2011.32020