research article black holes and quantum...

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


[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

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Statistical MechanicsInternational Journal of


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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



119889119872 = 119879119889119878 119878 =119896119888

4ℏ119866119860 (15)


119875

and119860 = 1198731198972



119879 =119889119872

119889119878=4ℏ119866

1198961198972

119875119888

119889119872

119889119873 (16)





119889119872

119889119905= minus

120573

1198722 (17)


119875= 119872 We

can now deduce that

119889119872

119889119905=119898119875

119905119875

119872 = (119898119875

119905119875

) sdot 119905

(18)








119860 = 1198731198972

119901 (19)




]119865

rarr

120590 sdotrarr

nabla 120595 (119903) = 119864120595 (119903) (20)




119865is some





References





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119889119872 = 119879119889119878 119878 =119896119888

4ℏ119866119860 (15)


119875

and119860 = 1198731198972



119879 =119889119872

119889119878=4ℏ119866

1198961198972

119875119888

119889119872

119889119873 (16)





119889119872

119889119905= minus

120573

1198722 (17)


119875= 119872 We

can now deduce that

119889119872

119889119905=119898119875

119905119875

119872 = (119898119875

119905119875

) sdot 119905

(18)








119860 = 1198731198972

119901 (19)




]119865

rarr

120590 sdotrarr

nabla 120595 (119903) = 119864120595 (119903) (20)




119865is some





References





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article black holes and quantum...

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