Journal of Modern Physics, 2011, 2, 350-353
doi:10.4236/jmp.2011.25043 Published Online May 2011 (http://www.SciRP.org/journal/jmp)
Can a Massive Graviton be a Stable Particle
Andrew Beckwith
Department of Physic, Chongquing University, Chongqing, China
Received January 26, 2011; revised April 1, 2011; accepted April 3, 2011
Abstract
This document is based on a question asked in the Dark Side of the Universe 2010 conference in Leon,
Mexico, when a researcher from India asked the author about how to obtain a stability analysis of massive
gravitons. The answer to this question involves an extension of the usual Pauli_Fiertz Langrangian as written
by Ortin, with non-zero graviton mass contributing to a relationship between the trace of a revised GR
stress-energy tensor (assuming non-zero graviton mass), and the trace of a revised symmetric tensor times a
tiny mass for a 4 dimensional graviton. The resulting analysis makes use of Visser’s treatment of a stress
energy tensor, with experimental applications discussed in the resulting analysis. If the square of frequency
of a massive graviton is real valued and greater than zero, stability can be possibly confirmed
experimentally.
Keywords: Graviton Stability, Gr Stress-Energy Tensor, 4 Dimensional Graviton
1. Introduction
The supposition advanced in this article is that initial
relic energy flux is central to making predictions as to
entropyf , where
Sn
f
nis a “particle count” per phase
space “volume” in the beginning of inflation [1,2]. So is
f
n due to gravitons in near-relic conditions? If so, can the
gravitons carry information? Where inf
entropyf start of
Sn


is about the value of entropy information, which is
bits of “information” in line with Smoot’s [3]
talk at the Paris observatory. Having said that, a relevant
issue raised in DSU 2010 is: if gravitons with a small
mass are part of the bridge between
7
10
10
10
7
inf 10
entropyf start of
Sn
  initial informa-
10
10
tion bits, can one make a statement about necessary
conditions for “massive” graviton stability? The
conclusion is that stability of a massive graviton needs
the square of frequency to be positive real valued.
2. What can be Said about Massive Graviton
Stability? Necessary Conditions
This document looks at work presented by Maggiore [4],
which specifically delineated for non-zero graviton mass,
where and v, that a small rest
mass of the graviton is proportional to the trace of the
energy-momentum tensor of general relativity.

Trace
uv
uv uv
hh h  
2
32
graviton
mh

T


(1)
This document uses Visser’s [5] analysis of non-zero
graviton mass for both T and h. Equation (1) is used with
particle count
f
n as a way to present initial GW relic
inflation density using the definition given by Maggiore
[4] as a way to state that a particle count:
 
2
0
0
4
37
dlog
3.6 1 kHz10
f
gw
gwgw gw
cf
f
f
fh f
nf




 


 
 



(2)
where
f
n is the frequency-based numerical count of
gravitons per unit phase space. To do so, let us give the
reasons for using Visser’s [5] values for T and h, in
Equation (1).
While Maggiore’s [4] explanation, and his treatment
of gravitational wave density is very good, the problem
we have is that any relic conditions for GW involve
stochastic background, and also that many theorists have
relied upon turbulence and or other forms of plasma-
induced generation of shock waves, as stated by Duerrer
and Rinaldi [6] and others, looking at the electroweak
transition as a GW generator. If relic conditions can also
yield GW/graviton production, and the consequences

A. BECKWITH351
exist up to the present era, then the question of stability
of gravitons is even more essential. The author [1,2] uses a
modification of results from Alcubierre [7], as an energy
flux value for GW/gravitons, given by the Equation (3).

223
2
64 π
initial fluxrPlanckeffective
r
Ehnt






 (3)
The
f
n value obtained, was used to make a
relationship, using Ng’s [8] entropy counting algorithm
of roughly entropy . The author suggests that in
order to obtain entropyf from initial graviton
production, as a way to quantify
f
S
S
n
n
f
n, that a small mass
of the graviton can be assumed. A small mass graviton in
four dimensions only makes sense if it is a stable
construct. The remainder of this article will give specific
cases to provide criteria for stability for the low mass 4
dimensional graviton in obtaining his value of
entropyf tying in what Ng’s [8] entropy values as
given by formulations started by Ng [8] and resultant
information content present in the early universe. In
doing so, the author will address if the correspondence
principle and the closeness of the links to massless
formalism of the graviton is due to ‘t Hoofts [9,10]; an
idea of embedding QM within deterministic quantum
theory, involving an embedding of quantum physics
within a slightly “larger” highly non linear structure
[1,2].
Sn
2.1. Defining the Graviton Problem and Using
Visser’s (1998) Inputs into
uv
T
This section defines a graviton in terms of a dark matter
component, and an effective dark energy contribution.
To do this, we look at a modification of what was
presented by Maartens [11], which is written as Equation
(4) below. Here we are actually using

65
Graviton10 grams
n
n
mL

 (4)
Equation (4) is a simplification of what is written for a
Kaluza Klein particle, which in this case would read as
given by Sarkar [1,2]

22
65
(graviton)10 grams
n
n
mL




 (4a)
On the face of it, this assignment of a mass of about
grams for a 4 dimensional graviton, allowing for
grams violates all known
quantum mechanics, and is to be avoided. Numerous
authors, including Maggiore [4], have richly demonstrated
how adding a term to the Fiertz Lagrangian as written up
by Ortin [13] (2007) for gravitons, and assuming massive
gravitons, leads to results which appear to violate field
theory. Turning to the problem, we can examine what
inputs to Equation (1) can tell us about whether there are
grounds for Equation (4) and what this says about
measurement protocol for both GW and gravitons as
given in Equation (2). Visser (1998) came up with inputs
into the GR stress tensor and also for the perturbing term
uv which will be given below. We will use them in
conjunction with Equation (1) to perform a stability
analysis of the consequences of setting the value of
65
10
0G

65
raviton-4~ 10mD

h


65
Graviton-4~10mD
0

grams and from there discuss
how to used ‘t Hooft’s (2002, 2006) supposition of
deterministic QM, as an embedding of QFT, and more
could play a role if there are conditions for stability of


65
Graviton-4~10mD
0

grams.
2.2. Visser’s Treatment of the Stress Energy
Tensor of GR, and Its Applications
This section will derive stability conditions for the
graviton, if the graviton has a small rest mass. Visser [5]
stated a stress energy tensor treatment of gravitons along
the lines of a four dimensional matrix treatment of the
stress energy tensor as given by


65
0
2
10
exp
uv m
g
T
r GM
rr



0
22
Graviton-4 ~
4000
0000
0000
0000
Pg
mD
GM
l





 















 








 (5)
Furthermore, his version of uv uv uv
g
h

 can be
written as setting

2exp 2
g
uv uv
mr
GM
hV
r

V







 



(6)
If one adds in velocity “reduction” put in for speed of
gravitons as stated by Visser [5]
24
22
1g
g
g
mc
vc


 
(7)
As well as often setting

15MG r for reasons
which Visser [5] outlined, one can insert all this into
Equation (1) to obtain a real value for the square of
frequency > 0, i.e.

2224 11 0
g
mc A 





; (8)
2
222
2
1
1exp
6
exp
g
ggPg
g
mr
r
Amcl
mr
MG
r






 
 








 
  
 






(9)

A. BECKWITH 352
According to
g
wgwc

 by Kim [14], if the
square of frequency of a graviton, with mass, is > 0, and
real valued, it is likely that the graviton is stable, at least
with regards to perturbations. Kim’s [14] article is with
regards to Gravitons in brane/string theory, but it is
likely that the same dynamic for semi classical
representations of a graviton with mass.
Conditions permitting equation (8) to have positive
values - This section is to obtain sufficient conditions for
stability of a graviton. Looking at Equation (8) is the
same as analyzing how
2
222
2
1
1exp
6
exp1
g
g
gPg
g
mr
r
Amc l
mr
MG
r






 

 











 
 

 





(10)
That is, setting
2
222
1
0exp
6
exp1
g
g gPg
g
mr
r
mc l
mr
MG
r




 
 











 


 
 




(11)
Note that Visser [5] writes
, and a wavelength
meters. The two values, as well as
ascertaining when one can use
29
210 eV
g
m 
22
~6 10
g
 
38
210 nucleon m


~1 5MG r, with r the
usual distance from a graviton generating source, and M
the mass’ of an object which would be a graviton emitter
put severe restrictions as to the volume of space time
values for which r could be ascertained. If, however,
Equation (10) had, in most cases, a setting for which,
then in many cases, Equation (8) would hold
0exp 1
g
g
mr
r









7
. (12)
The author believes that such a configuration would be
naturally occurring in most generation of gravitons at, or
before the electro weak transition point in early
cosmology evolution. The author believes, that Equation
(12) would allow to predict a particle count behavior
along the lines of , which is put into
Equation (2) and has implications for what to look for in
stochastic GW generation.
6
10to 10
f
n
2.3. Revisiting Ng’s Counting Algorithm for
Entropy, and Graviton Mass
The wavelength for a graviton as may be chosen to do
such an information exchange would be part of a
graviton as being part of an information counting
algorithm. Namely argue that when taking the log, that
the 1N term drops out. As used by Ng [8]


3
~1 !
N
N
ZNV   (13)
This, according to Ng [8] leads to entropy of the
limiting value of, if




log N SZ will be modified by
having the following done, namely after his use of
quantum infinite statistics,


3
log5 / 2SN VN 



  (14)
Eventually, the author hopes to put on a sound
foundation what ‘t Hooft [9,10] is doing with respect to
deterministic quantum mechanics and equivalence
classes embedding quantum particle structures.
Furthermore, making a count of gravitons with
gravitons,, with Lloyd’s [15] formalism 7 ~10SN

3/4 7
ln 2#~ 10
total B
I Skoperations (15)
as implying at least one operation per unit graviton, with
gravitons being one unit of information, per produced
graviton. Note, Smoot [3] gave initial values of the
operations as


10
#
initially
operations ~10 (16)
The author’s work tends to support this value, and if
gravitons are indeed stable in initial conditions,
information exchange between a prior to a present
universe may become a topic of experimental invest-
tigation.
3. Conclusion
The author pursued this question, partly due to wishing
to determine if a non brane theory way to identify
graviton stability existed. The author was particularly
impressed with Visser’s [5] treatment of gravitons in the
context of both an alleged graviton wave length, and the
net slow down of gravitons, as referenced in Equation (7).
Note, that the treatment of Equation (6) above heavily
depends upon a small mass to the graviton very slightly
lowering the speed of graviton to just below the speed of
light. As the graviton mass is slight, the velocity of a spin
two graviton is ALMOST the speed of light. If Equation
(12) is verified in measurement, and there is a search
done for regions of space time for graviton production,
then the author hopes for a refinement and vetting
experimentally as to Duerrer’s and Rinaldi [6]
supposition of turbulence in the electro weak transition
being the major source for GW/graviton production in
early universe cosmology. In addition, it may give
experimental evidence for the use of Alcubierre’s [7]
(2008) expression of energy density as associated with

A. BECKWITH 353
gravitational waves.
4. Nomenclature - Definitions
65
(Graviton) 10
n
mnL

 grams is a Kaluza Klein
graviton mass expression, with a slight rest mass put in,
for four dimensions, of grams. The n is for nodes
in 5 dimensions, and L is the length (size) of a 5th
dimension.
65
10
f
n = numerical density of a group of gravitons, per
unit phase space “volume” at a given frequency, f.
g
wgwc

 is a ratio of gravity wave “density” per
unit volume of phase space, over a phase space volume.
effective here, in this case is the same as 
g
w Pauli_
Fierz Langrangian = classical stability and absence of
ghosts lead directly to the standard Fierz-Pauli
Lagrangian.

KK. = Kaluza-Klein. A model that seeks to unify the
two fundamental forces of gravitation and
electromagnetism In the case of this paper, it is for
particles obeying a unification of gravitation and
electromagnetism.
DM, DE = Dark Matter, and Dark energy. Non
baryonic matter in cosmology.
N
Z
= partition function, a concept usually from
statistical physics.

= wavelength of a “particle”. Frequently in
association with matter as a particle and a wave, i.e.
wave- particle duality of quantum mechanics.
5. Acknowledgements
The author wishes to thank Dr. Fangyu Li as well as
Stuart Allen, of international media associates whom freed
the author to think about physics, and get back to his
work. This work is supported in part by National Nature
Science Foundation of China grant No. 11075224.
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