Can a Massive Graviton be a Stable Particle

doi:10.4236/jmp.2011.25043

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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

E-mail: [email protected]

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

nis a “particle count” per phase

space “volume” in the beginning of inflation [1,2]. So is

n due to gravitons in near-relic conditions? If so, can the

gravitons carry information? Where inf

entropyf start of





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

10

inf 10

entropyf start of

  initial informa-

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

hh h





graviton







(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

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:

 

dlog

3.6 1 kHz10

gwgw gw

fh f







 





 











(2)

where

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

64 π

initial fluxrPlanckeffective

Ehnt













 (3)

The

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

S

n

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

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



Graviton10 grams



 (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]



(graviton)10 grams









 (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

10



raviton-4~ 10mD







Graviton-4~10mD



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





Graviton-4~10mD



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





exp

uv m

r GM







Graviton-4 ~

4000

0000









 





































 (5)

Furthermore, his version of uv uv uv



 can be

written as setting



2exp 2

uv uv

















 







(6)

If one adds in velocity “reduction” put in for speed of

gravitons as stated by Visser [5]





 

(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

mc A













; (8)

222

1exp

exp

ggPg

Amcl











 





















 



 

 











(9)

A. BECKWITH

352

According to

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

222

1exp

exp1

gPg

Amc l













 





























 





 









(10)

That is, setting

222

0exp

exp1

gPg

mc l



































 





 

 





(11)

Note that Visser [5] writes

, and a wavelength

meters. The two values, as well as

ascertaining when one can use

210 eV

m



~6 10





210 nucleon





~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



















. (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.

10to 10

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]



~1 !

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,





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





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

(Graviton) 10

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.

10

n = numerical density of a group of gravitons, per

unit phase space “volume” at a given frequency, f.

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 

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.

= 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.

6. References

[1] A. W. Beckwith, “Applications of Euclidian Snyder Geo-

metry to the Foundations of Space-Time Physics,”

Electronic Journal of Theoretical Physics, Vol. 7, No. 24,

2010, pp. 241-266.

[2] A. Beckwith, “Energy Content of Gravitation as a Way to

Quantify Both Entropy and Information Generation in the

Early Universe,” Journal of Modern Physics, Vol. 2, No.

2, 2011, pp. 58-61. doi:10.4236/jmp.2011.22010

[3] G. Smoot, “CMB Observations and the Standard Model

of the Universe,” International Programme of Cosmology

Daniel Chalonge “Third Millennium,” Paris, 2007.

[4] M. Maggiore, “Gravitational Waves, Volume 1: Theory

and Experiment,” Oxford University Press, Oxford, 2008.

[5] M. Visser, “Mass for the Graviton,” General Relativity

and Gravitation, Vol. 30, No. 12, 1998, pp. 1717-1728.

doi:10.1023/A:1026611026766

[6] R. Durrer and M. Rinaldi, “Graviton Production in

Noninflationary Cosmology,” Physical Review D, Vol. 79,

No. 6, 2009, p. 063507. doi:10.1103/PhysRevD.79.063507

[7] M. Alcubierre, “Introduction to Numerical Relativity,”

Oxford University Press, Oxford, 2008.

[8] Y. Ng, “Spacetime Foam: From Entropy and Holography

to Infinite Statistics and Nonlocality,” Entropy 2008, Vol.

10, No. 4, 2008, pp. 441-461. doi:10.3390/e10040441

[9] G. ‘t Hooft, “The Mathematical Basis for Deterministic

Quantum Mechanics,” In: Th. M. Nieuwenhuizen, et al.,

Eds., Beyond the Quantum, World Scientific Publishing,

Singapore, 2006.

[10] G. ‘t Hooft, “Determinism beneath Quantum Mechanics,”

Report Number: ITP-02/69; SPIN-2002/45.

http://arxiv.org/abs/quant-ph/0212095

[11] R. Maartens, “Braneworld Cosmology, Chapter 7 - The

Physics of the Early Universe,” Lecture Notes in Physics,

Vol. 653, 2005, pp. 213-252.

[12] U. Sarkar, “Particle and Astroparticle Physics in ‘Series

of High Energy Physics, Cosmology and Gravitation’,”

Taylor & Francis, New York, 2008.

[13] T. Ortin, “Gravity and Strings,” Cambridge University

Press, Oxford, 2007.

[14] J. Y. Kim, “Stability and Fluctuation Modes of Giant

Gravitons with NSNS B Field,” Physics Letters B, Vol.

529, No. 1-2, 2002, pp. 150-162.

doi:10.1016/S0370-2693(02)01233-9

[15] Lloyd, S., “Computational Capacity of the Universe,”

Physical Review Letters, Vol. 88, No. 23, 2002, p.

237901. doi:10.1103/PhysRevLett.88.237901