Research ArticleBlack Holes and Quantum Mechanics
B G Sidharth12
1 International Institute for Applicable Mathematics amp Information Sciences Udine Italy2 BM Birla Science Centre Adarsh Nagar Hyderabad 500 063 India
Correspondence should be addressed to B G Sidharth iiamisbgsyahoocoin
Received 27 January 2014 Accepted 23 February 2014 Published 20 March 2014
Academic Editor Christian Corda
Copyright copy 2014 B G Sidharth This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3
We look at black holes from different novel perspectives
1 Introduction
We will first show that black holes generally thought to bea general relativistic phenomena could also be understoodwithout invoking general relativity at all (Indeed Laplace hadanticipated these objects)
We start by defining a black hole as an object at the surfaceof which the escape velocity equals the maximum possiblevelocity in the universe namely the velocity of light We nextuse the well-known equation of Keplerian orbits [1]
1
119903=119866119872
1198712(1 + 119890 cos 120579) (1)
where119871 the so-called impact parameter is given by119877119888 where119877 is the point of closest approach in our case a point on thesurface of the object and 119888 is the velocity of approach in ourcase the velocity of light
Choosing 120579 = 0 and 119890 asymp 1 we can deduce from (1)
119877 =2119866119872
1198882
(2)
Equation (2) gives the Schwarzschild radius for a black holeand can be deduced from the full general relativity theory aswell
Wewill nowuse (2) to exhibit black holes at three differentscales the micro- the macro- and the cosmic scales
2 Black Holes
Our starting point is the observation that a Planck mass10minus5 gms at the Planck length 10
minus33 cms satisfies (2) andas such a Schwarzschild black hole is Rosen has usednonrelativistic quantum theory to show that such a particleis a mini universe [2]
We next come to stellar scales It is well known that for anelectron gas in a highly dense mass we have [3 4]
119870(11987243
1198774
minus11987223
1198772) = 119870
10158401198722
1198774 (3)
where
(119870
1198701015840) = (
27120587
64120572)(
ℏ119888
1205741198982
119875
) asymp 1040 (4)
119872 =9120587
8
119872
119898119875
119877 =119877
(ℏ119898119890119888) (5)
119872 is the mass 119877 the radius of the body 119898119875and 119898
119890are
the proton and electron masses and ℏ is the reduced Planckconstant From (3) and (4) it is easy to see that for 119872 lt
1060 there are highly condensed planet sized stars (In fact
these considerations lead to theChandrasekhar limit in stellartheory) We can also verify that for 119872 approaching 10
60
corresponding to a mass sim1036 gms or roughly a hundred toa thousand times the solarmass the radius119877 gets smaller and
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 606439 4 pageshttpdxdoiorg1011552014606439
2 Advances in High Energy Physics
smaller and would be sim108 cms so as to satisfy (2) and give ablack hole in broad agreement with theory and observation
Finally for the universe as a whole using only the theoryof Newtonian gravitation we had deduced [5]
119877 sim2119866119872
1198882
(6)
that is (2) where this time 119877 sim 1028 cms is the radius of
the universe and 119872 sim 1055 gms is the mass of the universe
(6) can be deduced alternatively from general relativisticconsiderations also as noted
Equation (6) is the same as (2) and suggests that theuniverse itself is a black hole (This will still be true if thereis dark matter)
It is remarkable that if we consider the universe to be aSchwarzschild black hole as suggested by (6) the time takenby a ray of light to traverse the universe that is from thehorizon to the singularity namely 10minus5(119872119872
0) equals the
age of the universe sim1017 secs as shown elsewhere [5] 1198720is
the mass of the sum We will deduce this result alternativelya little later
3 Micro Black Holes
Attempts have been made to express elementary particlesas tiny black holes by several authors notably Markov andRecami [6 7] These black holes do not reproduce charge orspin which are so essential
Let us instead observe that if we treat an electron as aKerr-Newman black hole then we get the correct quantummechanical 119892 = 2 factor but the horizon of the black holebecomes complex [4 8] Consider
119903+=119866119872
1198882
+ 120484119887 119887 equiv (11986621198722
1198884
minus1198661198762
1198884
minus 1198862)
12
(7)
with 119866 being the gravitational constant 119872 being the massand 119886 equiv 119871119872119888 119871 being the angular momentum While(7) exhibits a naked singularity and as such has no physicalmeaning we note that from the realm of quantummechanicsthe position coordinate for a Dirac particle is given by
119909 = (11988821199011119867minus1119905) +
120484
2119888ℏ (1205721minus 1198881199011119867minus1)119867minus1 (8)
an expression that is very similar to (7) In the above thevarious symbols have their usual meaning In fact as wasargued in detail [4] the imaginary parts of both (7) and (8)are the same being of the order of the Compton wavelength
It is at this stage that a proper physical interpretationbegins to emerge Dirac himself observed that to interpret(8) meaningfully it must be remembered that quantummechanical measurements (unlike classical ones) are reallyaveraged over the Compton scale Within the scale there arethe unphysical Zitterbewegung effects for a point electron thevelocity equals that of light
Once such a minimum spacetime scale is invoked thenwe have a noncommutative geometry as shown by Snyder [910]
[119909 119910] = (1204841198862
ℏ)119871119911 [119905 119909] = (
1204841198862
ℏ119888)119872119909 etc
[119909 119901119909] = 120484ℏ [1 + (
119886
ℏ)
2
1199012
119909]
(9)
The relations (9) are compatible with special relativityIndeed such minimum spacetime models were studiedfor several decades precisely to overcome the divergencesencountered in quantum field theory [4 10ndash13]
All this is symptomatic of the fact that we cannotmeasurearbitrary small intervals of spacetime in quantum theory asindeed argued by Dirac himself [14] Indeed subsequentlySalecker and Wigner argued that time within the Comptonscale has no physical meaning [15] (and for a detaileddiscussion cf [16]) Indeed this quantum mechanical featureexplains what Misner et al termed the greatest crisis ofphysics [8] namely the singularity of the black hole All thishas been the matter of detailed study (cf [16])
4 Black Hole Thermodynamics
The author has approached this problem from the point ofview of oscillations at the Planck scale [16] Briefly if thereare119873 such oscillators with an amplitude Δ119909 then we have
119877 = radic119873Δ1199092 (10)
This leads to
119877 = radic119873119897119875 119872 =
119898119875
radic119873
(11)
where119872 is the arbitrarymass119877 the extent and 119897119875and119898
119875are
the Planck length and Planck mass respectively We now usethe fact that 119897
119875is the Schwarzschild radius of the Planck mass
as was shown by Rosen [2] Substitution in the above gives usthe Schwarzschild radius that is (4)
119877 =2119866119872
1198882
(12)
It can be immediately seen from (11) that
119877119872 = 119897119875119898119875 (13)
It must be mentioned that the above is completely consistentwith the mass and radius of an arbitrary black hole includingthe universe itself
From the theory of black hole thermodynamics we haveas it is well known [17]
119879 =ℏ1198883
8120587119896119898119866 (14)
namely the Beckenstein temperature Interestingly (14) canbe deduced alternatively fromour above theory of oscillations
Advances in High Energy Physics 3
at the Planck scale For this we use the following relations fora Schwarzschild black hole [17]
119889119872 = 119879119889119878 119878 =119896119888
4ℏ119866119860 (15)
where 119879 is the Bekenstein temperature 119878 the entropy and 119860is the area of the black hole In our case themass119872 = radic119873119898
119875
and119860 = 1198731198972
119875 where119873 is arbitrary for an arbitrary black hole
This follows from (11) Whence
119879 =119889119872
119889119878=4ℏ119866
1198961198972
119875119888
119889119872
119889119873 (16)
If we use the fact that 119897119875is the Schwarzschild radius for the
Planck mass 119898119875and use the expression for 119872 the above
reduces to (14) the Bekenstein formulaEquation (14) gives also the thermodynamic temperature
of a Planck mass black hole Further in this theory as it isknown [17]
119889119872
119889119905= minus
120573
1198722 (17)
with 119872 being the mass Before proceeding we observe thatwe have deduced a string of119873 Planck oscillators119873 arbitraryform a Schwarzschild black hole of mass radic119873119898
119875= 119872 We
can now deduce that
119889119872
119889119905=119898119875
119905119875
119872 = (119898119875
119905119875
) sdot 119905
(18)
where 119905 is the ldquoHawking-Bekenstein decay timerdquo For thePlanck mass 119872 = 119898
119875 the decay time is the Planck time
119905 = 119905119875 For the universe the above gives the life time 119905 as
sim1017 sec the age of the universe againFurther we have also seen the emergence of the quantum
of area [18] as it is evident from the119873 elementary Planck areas1198972
119875for the black hole (cf also [18])It has also been argued that not only does the universe
mimic a black hole but also the black hole is a two dimen-sional object [16 19] Indeed the interior of a black hole is inany case inaccessible and the two dimensions follow from thearea of the black hole which plays a central role in black holethermodynamics We have already seen that the area of theblack hole is given by
119860 = 1198731198972
119901 (19)
For these quantum gravity considerations we have to dealwith the quantum of area [16 18] In other words we haveto consider the black hole to be made up of119873 quanta of areaIt is remarkable that we can get an opportunity to test thesequantum gravity features in two-dimensional surfaces suchas graphene
That is we could model a black hole as a ldquographenerdquo ballIndeed in the case of graphene as it is well known and as
the author deduced in 1995 [20 21] this behaviour in twodimensions is given by
]119865
rarr
120590 sdotrarr
nabla 120595 (119903) = 119864120595 (119903) (20)
where ]119865sim 106ms is the Fermi velocity replacing 119888 the
velocity of light and 120595(119903) is a two-component wave functionrarr
120590 and 119864 denoting the Pauli matrices and energyThough this resembles the neutrino equation ]
119865is some
three hundred times less than the velocity of light Howeverthe author has argued that for a sufficiently large sheet ofgraphene this would approximate the neutrino equationitself that is the usualMinkowski spacetime From this pointof view a black hole can be simulated by a ldquographene ballrdquo
It may be mentioned that very recently Hawking hasproposed rather shockingly that black holes may not haveevent horizons [22]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] HGoldsteinClassicalMechanics Addison-Wesley Press Read-ing Mass USA 1951
[2] N Rosen ldquoQuantum mechanics of a miniuniverserdquo Interna-tional Journal ofTheoretical Physics vol 32 no 8 pp 1435ndash14401993
[3] K Huang Statistical Mechanics John Wiley amp Sons New YorkNY USA 2nd edition 1987
[4] B G Sidharth Chaotic Universe From the Planck to the HubbleScale Nova Science New York NY USA 2001
[5] B G Sidharth ldquolsquoFluctuational cosmologyrsquo in quantummechan-ics and general relativityrdquo in Proceeding of the 8th MarcellGrossmann Meeting on General Relativity T Piran Ed WorldScientific Singapore 1999
[6] M A Markov Soviet Physics JETP vol 24 no 3 p 584 1967[7] M A Markov Commemoration Issue for the 30th Anniversary
of the MesonTheory by Dr H Yukawa Suppl of Progress ofThPhys 1965
[8] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman San Francisco Calif USA 1973
[9] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 pp 68ndash71 1947
[10] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947
[11] D R Finkelstein Quantum Relativity A Synthesis of the Ideasof Einstein and Heisenberg Texts and Monographs in PhysicsSpringer Berlin Germany 1996
[12] C Wolf HadronicJournal vol 13 pp 22ndash29 1990[13] T D Lee ldquoCan time be a discrete dynamical variable rdquo Physics
Letters vol 122B no 3-4 pp 217ndash220 1983[14] P AMDiracThePrinciples of QuantumMechanics Clarendon
Press Oxford UK 3rd edition 1947[15] H Salecker and E P Wigner ldquoQuantum limitations of the
measurement of space-time distancesrdquo Physical Review vol 109pp 571ndash577 1958
4 Advances in High Energy Physics
[16] B G SidharthTheThermodynamic Universe World ScientificSingapore 2008
[17] R Runi and L Z Zang Basic Concepts in Relativistic Astro-Physics World Scientific Singapore 1983
[18] J Baez ldquoQuantum gravity the quantum of areardquo Nature vol421 no 6924 pp 702ndash703 2003
[19] B G Sidharth ldquoBlack-hole thermodynamics and electromag-netismrdquo Foundations of Physics Letters vol 19 no 1 pp 87ndash942006
[20] B G Sidharth A Note on Two Dimensional Fermions in BSC-CAMCS-TR-95-04-01 1995
[21] B G Sidharth ldquoAnomalous fermionsrdquo Journal of StatisticalPhysics vol 95 no 3-4 pp 775ndash784 1999
[22] S W Hawking ldquoInformation preservation and weather fore-casting for black holesrdquo httparxivorgabs14015761
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
2 Advances in High Energy Physics
smaller and would be sim108 cms so as to satisfy (2) and give ablack hole in broad agreement with theory and observation
Finally for the universe as a whole using only the theoryof Newtonian gravitation we had deduced [5]
119877 sim2119866119872
1198882
(6)
that is (2) where this time 119877 sim 1028 cms is the radius of
the universe and 119872 sim 1055 gms is the mass of the universe
(6) can be deduced alternatively from general relativisticconsiderations also as noted
Equation (6) is the same as (2) and suggests that theuniverse itself is a black hole (This will still be true if thereis dark matter)
It is remarkable that if we consider the universe to be aSchwarzschild black hole as suggested by (6) the time takenby a ray of light to traverse the universe that is from thehorizon to the singularity namely 10minus5(119872119872
0) equals the
age of the universe sim1017 secs as shown elsewhere [5] 1198720is
the mass of the sum We will deduce this result alternativelya little later
3 Micro Black Holes
Attempts have been made to express elementary particlesas tiny black holes by several authors notably Markov andRecami [6 7] These black holes do not reproduce charge orspin which are so essential
Let us instead observe that if we treat an electron as aKerr-Newman black hole then we get the correct quantummechanical 119892 = 2 factor but the horizon of the black holebecomes complex [4 8] Consider
119903+=119866119872
1198882
+ 120484119887 119887 equiv (11986621198722
1198884
minus1198661198762
1198884
minus 1198862)
12
(7)
with 119866 being the gravitational constant 119872 being the massand 119886 equiv 119871119872119888 119871 being the angular momentum While(7) exhibits a naked singularity and as such has no physicalmeaning we note that from the realm of quantummechanicsthe position coordinate for a Dirac particle is given by
119909 = (11988821199011119867minus1119905) +
120484
2119888ℏ (1205721minus 1198881199011119867minus1)119867minus1 (8)
an expression that is very similar to (7) In the above thevarious symbols have their usual meaning In fact as wasargued in detail [4] the imaginary parts of both (7) and (8)are the same being of the order of the Compton wavelength
It is at this stage that a proper physical interpretationbegins to emerge Dirac himself observed that to interpret(8) meaningfully it must be remembered that quantummechanical measurements (unlike classical ones) are reallyaveraged over the Compton scale Within the scale there arethe unphysical Zitterbewegung effects for a point electron thevelocity equals that of light
Once such a minimum spacetime scale is invoked thenwe have a noncommutative geometry as shown by Snyder [910]
[119909 119910] = (1204841198862
ℏ)119871119911 [119905 119909] = (
1204841198862
ℏ119888)119872119909 etc
[119909 119901119909] = 120484ℏ [1 + (
119886
ℏ)
2
1199012
119909]
(9)
The relations (9) are compatible with special relativityIndeed such minimum spacetime models were studiedfor several decades precisely to overcome the divergencesencountered in quantum field theory [4 10ndash13]
All this is symptomatic of the fact that we cannotmeasurearbitrary small intervals of spacetime in quantum theory asindeed argued by Dirac himself [14] Indeed subsequentlySalecker and Wigner argued that time within the Comptonscale has no physical meaning [15] (and for a detaileddiscussion cf [16]) Indeed this quantum mechanical featureexplains what Misner et al termed the greatest crisis ofphysics [8] namely the singularity of the black hole All thishas been the matter of detailed study (cf [16])
4 Black Hole Thermodynamics
The author has approached this problem from the point ofview of oscillations at the Planck scale [16] Briefly if thereare119873 such oscillators with an amplitude Δ119909 then we have
119877 = radic119873Δ1199092 (10)
This leads to
119877 = radic119873119897119875 119872 =
119898119875
radic119873
(11)
where119872 is the arbitrarymass119877 the extent and 119897119875and119898
119875are
the Planck length and Planck mass respectively We now usethe fact that 119897
119875is the Schwarzschild radius of the Planck mass
as was shown by Rosen [2] Substitution in the above gives usthe Schwarzschild radius that is (4)
119877 =2119866119872
1198882
(12)
It can be immediately seen from (11) that
119877119872 = 119897119875119898119875 (13)
It must be mentioned that the above is completely consistentwith the mass and radius of an arbitrary black hole includingthe universe itself
From the theory of black hole thermodynamics we haveas it is well known [17]
119879 =ℏ1198883
8120587119896119898119866 (14)
namely the Beckenstein temperature Interestingly (14) canbe deduced alternatively fromour above theory of oscillations
Advances in High Energy Physics 3
at the Planck scale For this we use the following relations fora Schwarzschild black hole [17]
119889119872 = 119879119889119878 119878 =119896119888
4ℏ119866119860 (15)
where 119879 is the Bekenstein temperature 119878 the entropy and 119860is the area of the black hole In our case themass119872 = radic119873119898
119875
and119860 = 1198731198972
119875 where119873 is arbitrary for an arbitrary black hole
This follows from (11) Whence
119879 =119889119872
119889119878=4ℏ119866
1198961198972
119875119888
119889119872
119889119873 (16)
If we use the fact that 119897119875is the Schwarzschild radius for the
Planck mass 119898119875and use the expression for 119872 the above
reduces to (14) the Bekenstein formulaEquation (14) gives also the thermodynamic temperature
of a Planck mass black hole Further in this theory as it isknown [17]
119889119872
119889119905= minus
120573
1198722 (17)
with 119872 being the mass Before proceeding we observe thatwe have deduced a string of119873 Planck oscillators119873 arbitraryform a Schwarzschild black hole of mass radic119873119898
119875= 119872 We
can now deduce that
119889119872
119889119905=119898119875
119905119875
119872 = (119898119875
119905119875
) sdot 119905
(18)
where 119905 is the ldquoHawking-Bekenstein decay timerdquo For thePlanck mass 119872 = 119898
119875 the decay time is the Planck time
119905 = 119905119875 For the universe the above gives the life time 119905 as
sim1017 sec the age of the universe againFurther we have also seen the emergence of the quantum
of area [18] as it is evident from the119873 elementary Planck areas1198972
119875for the black hole (cf also [18])It has also been argued that not only does the universe
mimic a black hole but also the black hole is a two dimen-sional object [16 19] Indeed the interior of a black hole is inany case inaccessible and the two dimensions follow from thearea of the black hole which plays a central role in black holethermodynamics We have already seen that the area of theblack hole is given by
119860 = 1198731198972
119901 (19)
For these quantum gravity considerations we have to dealwith the quantum of area [16 18] In other words we haveto consider the black hole to be made up of119873 quanta of areaIt is remarkable that we can get an opportunity to test thesequantum gravity features in two-dimensional surfaces suchas graphene
That is we could model a black hole as a ldquographenerdquo ballIndeed in the case of graphene as it is well known and as
the author deduced in 1995 [20 21] this behaviour in twodimensions is given by
]119865
rarr
120590 sdotrarr
nabla 120595 (119903) = 119864120595 (119903) (20)
where ]119865sim 106ms is the Fermi velocity replacing 119888 the
velocity of light and 120595(119903) is a two-component wave functionrarr
120590 and 119864 denoting the Pauli matrices and energyThough this resembles the neutrino equation ]
119865is some
three hundred times less than the velocity of light Howeverthe author has argued that for a sufficiently large sheet ofgraphene this would approximate the neutrino equationitself that is the usualMinkowski spacetime From this pointof view a black hole can be simulated by a ldquographene ballrdquo
It may be mentioned that very recently Hawking hasproposed rather shockingly that black holes may not haveevent horizons [22]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] HGoldsteinClassicalMechanics Addison-Wesley Press Read-ing Mass USA 1951
[2] N Rosen ldquoQuantum mechanics of a miniuniverserdquo Interna-tional Journal ofTheoretical Physics vol 32 no 8 pp 1435ndash14401993
[3] K Huang Statistical Mechanics John Wiley amp Sons New YorkNY USA 2nd edition 1987
[4] B G Sidharth Chaotic Universe From the Planck to the HubbleScale Nova Science New York NY USA 2001
[5] B G Sidharth ldquolsquoFluctuational cosmologyrsquo in quantummechan-ics and general relativityrdquo in Proceeding of the 8th MarcellGrossmann Meeting on General Relativity T Piran Ed WorldScientific Singapore 1999
[6] M A Markov Soviet Physics JETP vol 24 no 3 p 584 1967[7] M A Markov Commemoration Issue for the 30th Anniversary
of the MesonTheory by Dr H Yukawa Suppl of Progress ofThPhys 1965
[8] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman San Francisco Calif USA 1973
[9] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 pp 68ndash71 1947
[10] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947
[11] D R Finkelstein Quantum Relativity A Synthesis of the Ideasof Einstein and Heisenberg Texts and Monographs in PhysicsSpringer Berlin Germany 1996
[12] C Wolf HadronicJournal vol 13 pp 22ndash29 1990[13] T D Lee ldquoCan time be a discrete dynamical variable rdquo Physics
Letters vol 122B no 3-4 pp 217ndash220 1983[14] P AMDiracThePrinciples of QuantumMechanics Clarendon
Press Oxford UK 3rd edition 1947[15] H Salecker and E P Wigner ldquoQuantum limitations of the
measurement of space-time distancesrdquo Physical Review vol 109pp 571ndash577 1958
4 Advances in High Energy Physics
[16] B G SidharthTheThermodynamic Universe World ScientificSingapore 2008
[17] R Runi and L Z Zang Basic Concepts in Relativistic Astro-Physics World Scientific Singapore 1983
[18] J Baez ldquoQuantum gravity the quantum of areardquo Nature vol421 no 6924 pp 702ndash703 2003
[19] B G Sidharth ldquoBlack-hole thermodynamics and electromag-netismrdquo Foundations of Physics Letters vol 19 no 1 pp 87ndash942006
[20] B G Sidharth A Note on Two Dimensional Fermions in BSC-CAMCS-TR-95-04-01 1995
[21] B G Sidharth ldquoAnomalous fermionsrdquo Journal of StatisticalPhysics vol 95 no 3-4 pp 775ndash784 1999
[22] S W Hawking ldquoInformation preservation and weather fore-casting for black holesrdquo httparxivorgabs14015761
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 3
at the Planck scale For this we use the following relations fora Schwarzschild black hole [17]
119889119872 = 119879119889119878 119878 =119896119888
4ℏ119866119860 (15)
where 119879 is the Bekenstein temperature 119878 the entropy and 119860is the area of the black hole In our case themass119872 = radic119873119898
119875
and119860 = 1198731198972
119875 where119873 is arbitrary for an arbitrary black hole
This follows from (11) Whence
119879 =119889119872
119889119878=4ℏ119866
1198961198972
119875119888
119889119872
119889119873 (16)
If we use the fact that 119897119875is the Schwarzschild radius for the
Planck mass 119898119875and use the expression for 119872 the above
reduces to (14) the Bekenstein formulaEquation (14) gives also the thermodynamic temperature
of a Planck mass black hole Further in this theory as it isknown [17]
119889119872
119889119905= minus
120573
1198722 (17)
with 119872 being the mass Before proceeding we observe thatwe have deduced a string of119873 Planck oscillators119873 arbitraryform a Schwarzschild black hole of mass radic119873119898
119875= 119872 We
can now deduce that
119889119872
119889119905=119898119875
119905119875
119872 = (119898119875
119905119875
) sdot 119905
(18)
where 119905 is the ldquoHawking-Bekenstein decay timerdquo For thePlanck mass 119872 = 119898
119875 the decay time is the Planck time
119905 = 119905119875 For the universe the above gives the life time 119905 as
sim1017 sec the age of the universe againFurther we have also seen the emergence of the quantum
of area [18] as it is evident from the119873 elementary Planck areas1198972
119875for the black hole (cf also [18])It has also been argued that not only does the universe
mimic a black hole but also the black hole is a two dimen-sional object [16 19] Indeed the interior of a black hole is inany case inaccessible and the two dimensions follow from thearea of the black hole which plays a central role in black holethermodynamics We have already seen that the area of theblack hole is given by
119860 = 1198731198972
119901 (19)
For these quantum gravity considerations we have to dealwith the quantum of area [16 18] In other words we haveto consider the black hole to be made up of119873 quanta of areaIt is remarkable that we can get an opportunity to test thesequantum gravity features in two-dimensional surfaces suchas graphene
That is we could model a black hole as a ldquographenerdquo ballIndeed in the case of graphene as it is well known and as
the author deduced in 1995 [20 21] this behaviour in twodimensions is given by
]119865
rarr
120590 sdotrarr
nabla 120595 (119903) = 119864120595 (119903) (20)
where ]119865sim 106ms is the Fermi velocity replacing 119888 the
velocity of light and 120595(119903) is a two-component wave functionrarr
120590 and 119864 denoting the Pauli matrices and energyThough this resembles the neutrino equation ]
119865is some
three hundred times less than the velocity of light Howeverthe author has argued that for a sufficiently large sheet ofgraphene this would approximate the neutrino equationitself that is the usualMinkowski spacetime From this pointof view a black hole can be simulated by a ldquographene ballrdquo
It may be mentioned that very recently Hawking hasproposed rather shockingly that black holes may not haveevent horizons [22]
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] HGoldsteinClassicalMechanics Addison-Wesley Press Read-ing Mass USA 1951
[2] N Rosen ldquoQuantum mechanics of a miniuniverserdquo Interna-tional Journal ofTheoretical Physics vol 32 no 8 pp 1435ndash14401993
[3] K Huang Statistical Mechanics John Wiley amp Sons New YorkNY USA 2nd edition 1987
[4] B G Sidharth Chaotic Universe From the Planck to the HubbleScale Nova Science New York NY USA 2001
[5] B G Sidharth ldquolsquoFluctuational cosmologyrsquo in quantummechan-ics and general relativityrdquo in Proceeding of the 8th MarcellGrossmann Meeting on General Relativity T Piran Ed WorldScientific Singapore 1999
[6] M A Markov Soviet Physics JETP vol 24 no 3 p 584 1967[7] M A Markov Commemoration Issue for the 30th Anniversary
of the MesonTheory by Dr H Yukawa Suppl of Progress ofThPhys 1965
[8] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman San Francisco Calif USA 1973
[9] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 pp 68ndash71 1947
[10] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947
[11] D R Finkelstein Quantum Relativity A Synthesis of the Ideasof Einstein and Heisenberg Texts and Monographs in PhysicsSpringer Berlin Germany 1996
[12] C Wolf HadronicJournal vol 13 pp 22ndash29 1990[13] T D Lee ldquoCan time be a discrete dynamical variable rdquo Physics
Letters vol 122B no 3-4 pp 217ndash220 1983[14] P AMDiracThePrinciples of QuantumMechanics Clarendon
Press Oxford UK 3rd edition 1947[15] H Salecker and E P Wigner ldquoQuantum limitations of the
measurement of space-time distancesrdquo Physical Review vol 109pp 571ndash577 1958
4 Advances in High Energy Physics
[16] B G SidharthTheThermodynamic Universe World ScientificSingapore 2008
[17] R Runi and L Z Zang Basic Concepts in Relativistic Astro-Physics World Scientific Singapore 1983
[18] J Baez ldquoQuantum gravity the quantum of areardquo Nature vol421 no 6924 pp 702ndash703 2003
[19] B G Sidharth ldquoBlack-hole thermodynamics and electromag-netismrdquo Foundations of Physics Letters vol 19 no 1 pp 87ndash942006
[20] B G Sidharth A Note on Two Dimensional Fermions in BSC-CAMCS-TR-95-04-01 1995
[21] B G Sidharth ldquoAnomalous fermionsrdquo Journal of StatisticalPhysics vol 95 no 3-4 pp 775ndash784 1999
[22] S W Hawking ldquoInformation preservation and weather fore-casting for black holesrdquo httparxivorgabs14015761
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Advances in High Energy Physics
[16] B G SidharthTheThermodynamic Universe World ScientificSingapore 2008
[17] R Runi and L Z Zang Basic Concepts in Relativistic Astro-Physics World Scientific Singapore 1983
[18] J Baez ldquoQuantum gravity the quantum of areardquo Nature vol421 no 6924 pp 702ndash703 2003
[19] B G Sidharth ldquoBlack-hole thermodynamics and electromag-netismrdquo Foundations of Physics Letters vol 19 no 1 pp 87ndash942006
[20] B G Sidharth A Note on Two Dimensional Fermions in BSC-CAMCS-TR-95-04-01 1995
[21] B G Sidharth ldquoAnomalous fermionsrdquo Journal of StatisticalPhysics vol 95 no 3-4 pp 775ndash784 1999
[22] S W Hawking ldquoInformation preservation and weather fore-casting for black holesrdquo httparxivorgabs14015761
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of