Frenet-Serret vacuum radiation, detection proposals and related topics

28 Pages·0.23 MB·English

by  H. C. Rosu

1
FRENET-SERRET VACUUMRADIATION,DETECTION PROPOSALS
ANDRELATED TOPICS
Haret C. Rosu1‡
1 Dept. ofAppliedMathematics
PotosinianInstituteofScientificand TechnologicalResearch
ApdoPostal 3-74Tangamanga,San LuisPotos´ı,MEXICO
Dated: January 2003
3
0
0 Abstract
2 The paradigmatic Unruh radiation is an ideal and simplecase of stationary vacuum radiation

n patterns related to worldlines defined as Frenet-Serret curves. We review the corresponding
a body of literature as well as the experimental proposals that have been suggested to detect
J
these types of quantum field radiation patterns. Finally, we comment on a few other topics
7
1 related totheUnruh effect.

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v
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1 Part 1. Frenet-Serret Vacuum Radiation: pp. 2-11
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/ Part 2. DetectionProposals: pp. 12-22
h
t
-
p
e Part 3. Related Topics: pp. 23-26
h
:
v
i References: pp. 27-28
X
r
a
extended versionof invitedtalkat QABP3,Hiroshima,Japan,January7-11,2003
‡ [email protected]
2
1. FRENET-SERRET VACUUMRADIATION
J.F. Frenet (1816-1900) wrote a doctoral thesis in 1847. Part of it deals with the theory of
space curves and containssix formulas oftheninethree-dimensionalFrenet-Serret formulas.
He publishedhis results in Journalde mathematiquespures et appliques in 1852. J.A. Serret
(1819-1885)gavethesetofall nineformulasin threedimensions.
ThestandardFS formulasare thefollowing
˙
~T =k ~N
˙ ·
~N =t ~B k ~T
· − ·
˙
~B = t ~N .
− ·
These three formulas give the derivatives of the unit tangent, normal, and binormal vectors,
respectively, as vectorial combinations of the moving basis in which the two Frenet-Serret
invariantsofthenormalthree-dimensionalspace, curvatureand torsion,respectively,enteras
scalarcoefficients. Theycan bewrittenin matrixformas well
˙
~T 0 k 0 ~T
~N˙ = k 0 t ~N
 
~B˙  −0 t 0  ~B 
−
 
  
 
The plane formed by the span of~T and ~N is called the osculating plane. The span of~N and
~B is thenormalplane and thespan of~B and~T isthetransverseplaneto thecurveat thepoint
wheretheFrenet-Serret triad isconsidered.
For the goals of the first part of this review, the interested reader may consult several papers
of more recent times. These are seminal papers that apply the Frenet-Serret formalism to
relativistic world lines and quantum field theory. First, as quoted by J.L. Synge, Jasper [1]
studied in 1947 helices in flat space of n dimensions, but only for a positive-definite metric.
Next, Synge wrote an important paper on timelike helices in flat space-time and classified
them in six types according to the relative magnitudes of the three Frenet-Serret invariants
[2]. He also proved for the first time that the world line of a charged particle moving in a
constant electromagnetic field is a helix. A few years later, in the seminal paper of Honig,
SchuckingandVishveshwara[3]aconnectionwasestablishedbetweentheLorentzinvariants
of the elctromagneticfield and the Frenet-Serret invariants. The 1981 Letaw’s paper [4] is in
fact thefirst work inwhichtheFS invariantsand quantumfield conceptsare merged together
(see below). Interesting generalizations to more dimensions, black hole environments, and
gyroscopicprecessionhavebeen investigatedby Iyerand Vishveshwara[5](seealso [6]).
We also notice that at the strictly classical level, the Frenet-Serret coordinate system is quite
used inaccelerator physicstoexpress Hamiltoniansand/orfields [7].
3
1.1. Stationary worldlinesand thevacuum excitationofnoninertial detectors
In 1981, Letaw studied the stationary world lines, on which quantized field detectors in
a scalar vacuum have time-independent excitation spectra. They are characterized by the
requirement that the geodetic interval between two points depends only on the proper time
interval. He used a generalization of the Frenet equations to Minkowski space and found,
not surprisingly,that thecurvatureinvariantsare the properacceleration and angular velocity
of the world line. Solving the generalized Frenet equations for the simple case of constant
invariants leads to several classes of stationary world lines. Letaw gave a classification into
six types reviewed here. He also demonstrated the equivalenceof the timelike Killing vector
field orbits and the stationary world lines. Last but not least, Letaw did some calculations of
the vacuum excitation spectra of detectors on the sample of six families of stationary world
lines. Letaw’s work is a generalization of Unruh’s famous result concerning the excitation
of a scalar-particle detector moving with constant linear acceleration in the vacuum of flat
spacetime. According to Unruh, the detector behaves as if in contact with a bath of scalar
particles with energies in a Planck spectrum of temperature: acceleration/2p . Because of the
connectionwiththeHawkingradiationanditsparadigmaticnature,theUnruheffectattracted
the attention of many theoretical physicists. On the other hand, the present author considers
that Letaw’s results place the Unruh effect in a different and more general perspective that
is elaborated herein. We hope that a more effective attitude is put forward in the present
review that could be more rewarding in the years to come and may help to strengthen the
linksbetween laboratoryphysicsand astrophysics.
According to DeWitt, the probability for a detector moving along a world line xm (t ) to be
foundinanexcitedstateofenergyE att =t isgivenintermsoftheautocorrelationfunction
0
ofthefield (Wightmanfunction)
t t
P(E)=D(E) 0dt 0dt ′e iE(t t ′) 0 f (x(t ))f (x(t ′)) 0 , (1)
− −
¥ ¥ h | | i
Z− Z−
whereD(E)isafunctioncharacterizingthesensitivityofthedetector. TheWightmanfunction
forascalarfield is proportionalto theinverseofthegeodeticintervalD (t ,t ′)
0 f (x(t ))f (x(t ′)) 0 =[2p 2D (t ,t ′)] 1 , (2)
−
h | | i
where
D (t ,t ′)=[xm (t ) xm (t ′)][xm (t ) xm (t ′)] . (3)
− −
Therateofexcitationto thestatewithenergy E is theFouriercosinetransform
dP(E) 0
=2D(E) ds 0 f (x(t ))f (x(t +s)) 0 cos(Es). (4)
dt 0 ¥ h | 0 0 | i
Z−
where s=t t ′ is the proper time interval. The rate of excitation (the response) is directly
−
related totheenergy spectrumofthedetected scalar“particles”
0
S(E,t )=2pr (E) ds 0 f (x(t ))f (x(t +s)) 0 cos(Es). (5)
¥ h | | i
Z−
If the Wightman function is time independent the detected spectra are stationary. This is
equivalenttothefollowingpropertyofthegeodeticinterval
D (t ,t +s)=D (0,s). (6)
4
CurvatureinvariantsandtheFrenetequations
m
An arbitrary timelike world line in flat space is generally described by four functions, x (s),
specifyingthecoordinatesofeachpointsonthecurve. Thisparametermaybetakentobethe
arclengthorpropertimeontheworldline. Theparametricrepresentationisunsatisfactoryin
tworespects:
(1) A world line is a geometric object and should not require a coordinate-dependent entity
foritsdefinition.
(2) There is an inherent redundancy in the parametric representation since three functions
sufficetodeterminetheworldline.
The Frenet-Serret (curvature) invariants, on the other hand, provide an intrinsic definition of
theworldlinenot subjecttothesecriticisms.
m
The starting point is the construction of an orthonormal tetrad V (s) at every point on the
a
m
world line x (s). The Latin index everywhere is a tetrad index. The tetrad is formed from
m
the derivatives of x (s) with respect to proper time (represented by one or more dots). It is
assumed that the first four derivatives are linearly independent, the results being practically
unchangedwhentheyarenot. Membersofthetetradmustsatisfytheorthonormalitycondition
m
Vam Vb =h ab , (7)
wherethemetrichas diagonalcomponents(1,-1,-1,-1)only.
By Gram-Schmidt orthogonalization of the derivatives working upwords from the first,
thefollowingexpressionsforthetetrad are found:
m m
V =x˙ , (8)
0
a
m x¨
V = , (9)
1 ( x¨a x¨a )1/2
−
m (x¨g x¨g ).x..m (x¨g .x..g )x¨m +(x¨g x¨g )2x˙m
V = − , (10)
2 [(x¨a x¨a )4+(x¨a x¨a )(x¨b .x..b )2 (x¨a x¨a )2(.x..b .x..b )]1/2
−
V3m = √16e abgm V0a V1b V2g , (11)
Overallsignsonthesevectorsare fixedby theorientationofthetetrad.
m
The tetradVs is a basis for the vector space at a point on the world line. Derivatives of
thebasisvectorsmaytherefore beexpandedinterms ofthem:
V˙m =KbVm . (12)
a a b
These are the generalized Frenet equations. K is a coordinate-independent matrix whose
ab
structuremustbedetermined.
Differentiationoftheorthonormalitycondition(7)yields
m m
V˙am Vb +Vam V˙b =0 , (13)
and, inviewof(12),
K = K . (14)
ab ba
−
5
A basisvectorVm isdefined in termsofthefirst a+1derivativesofxm ; therefore,V˙m willbe
a a
a linearcombinationofthe first a+2 derivatives. Thesea+2 derivativesare dependent only
m
on thebasisvectorsV whereb a+1. This and(14)limitthematrixtotheform
b ≤
0 k (s) 0 0
k (s) − 0 t (s) 0
Kab = 0 t − 01 t  (15)
1 2
0 0 t −0
 2 
 
Thethreefunctionsofpropertimeare theinvariants
m m
k =V0m V˙1 =−V˙0m V1 , (16)
m m
t 1 =V1m V˙2 =−V˙1m V2 , (17)
m m
t 2 =V2m V˙3 =−V˙2m V3 . (18)
They are, respectively, the curvature, the first torsion, and the second torsion (hypertorsion)
oftheworldline. Sign choicesare madeforreasonsbroughtoutbelow.
To explore the physical significance of the invariants we examine the infinitesimal
Lorentz transformations of the tetrad at a point on the world line. The transformations leave
themetricinvariant
h =LcLdh . (19)
ab a b cd
An infinitesimaltransformationsmaybewritten
Lc =d c+de c , (20)
a a a
wheretheelementsofde c are smalland mustsatisfy
a
de = de , (21)
ab ba
−
The transformations are taken to be active; that is, the transformed tetrad moves +v and is
rotated q relativetotheuntransformedtetrad. Thustheinfinitesimalgeneratoris
0 dv dv dv
1 2 3
dv −0 −dq −dq
de ab = dv1 dq − 0 12 dq31  (22)
2 12 23
dv dq dq − 0
 3 − 31 23 
 
Thechangeinthetetrad resultingfrom thistransformationis
V˙m =(de b/ds)Vm . (23)
a a b
Equations(23)areidenticaltotheFrenetequations(12);therefore,thephysicalcontentofthe
curvatureinvariantsisfound bycomparisonof(15)and(22).
PhysicalinterpretationoftheFS invariants
m
1. k is theproperaccelerationoftheworldlinewhich isalways paralleltoV .
1
2. t and t are the components of proper angular velocity of the world line in the planes
1 2
m m m m
spannedbyV andV ,andV andV ,respectively. Thetotalproperangularvelocityisthe
1 2 2 3
vector sumof thesetwo invariants.
6
StationaryMotions
The simplest worldlines are those whose curvature invariants are constant. They are called
stationarybecause their geometric properties are independent of proper time. One also finds
that only observers on these world lines may establish a coordinate system in which they are
at rest and the metric is stationary. Clearly, the geodetic interval between two points on a
stationaryworldlinecandependonlyonthepropertimeinterval,thereforetheyaretheworld
lineson whichadetector’sexcitationistimeindependent.
m
The Frenet equation (12) may be reduced to a fourth-order linear equation in V when the
0
curvatureinvariantsare constant
....m m m
V 2aV¨ b2V =0 , (24)
0 − 0 − 0
where
1
a= k 2 t 2 t 2 , b= kt .
2 − 1 − 2 | 2|
(cid:0) m (cid:1)
Theotherbasis vectorsaredeterminedfromV bytheequations
0
m m
V =V˙ /k (25)
1 0
m m m
V =(V¨ k 2V )/kt (26)
2 0 − 0 1
m ...m m
V =[V (k 2 t 2)V˙ ]/kt t . (27)
3 0 − − 1 0 1 2
Equation(24) is homogeneouswithconstant coefficients. The fourroots of thecharacteristic
equationare R and iR , where
+
± ± −
R =[(a2+b2)1/2 a]1/2 . (28)
± ±
Thesolutionisoftheform
m m m m m
V =A cosh(R s)+B sinh(R s)+C cos(R s)+D sin(R s). (29)
0 + +
− −
m m
Using (25)-(27) and (29) at s = 0 and the initial conditions (V ) = d one can get the
a s=0 a
followingexpressionsforthecoefficients
Am =R 2(R2 +k 2,0,kt ,0), (30)
− 1
Bm =R 2(0,−k (R2 +k 2 t 2)/R ,0,kt t /R ) , (31)
− + 1 2 +
−
Cm =R 2(R2 k−2,0, kt ,0), (32)
− + 1
− −
Dm =R 2(0,k (R2 k 2+t 2)/R ,0, kt t /R ) , (33)
− + 1 2
− − − −
(34)
with R2 = R2 +R2. From these results on the FS tetrad one gets easily the six classes of
+
stationaryworldlin−esand thecorrespondingexcitationspectra.
7
TheSixStationaryScalarFrenet-SerretRadiationSpectra
1. k =t =t =0 (inertial, uncurvedworldlines)
1 2
Theexcitationspectrumis atrivialcubicspectrum
E3
S (E)= , (35)
recta 4p 2
i.e., asgivenbyavacuumofzero pointenergy per modeE/2 anddensityofstatesE2/2p 2.
2. k =0, t =t =0 (hyperbolicworldlines)
1 2
6
TheexcitationspectrumisPlanckianallowingtheinterpretationofk /2p as‘thermodynamic’
temperature. Inthedimensionlessvariablee k =E/k thevacuumspectrumreads
e 3
Shyp(e k )= 2p 2(e2pek k 1) . (36)
−
3. k < t , t =0,r 2 =t 2 k 2 (helical worldlines)
| | | 1| 2 1 −
Theexcitationspectrumis an analyticfunctioncorrespondingto thecase 4 belowonly inthe
limitk r
≫
Shel(e r )k /r →¥ S3/2 parab(e k ) . (37)
−→ −
LetawplottedthenumericalintegralShel(e r ),where e r =E/r forvariousvaluesofk /r .
4. k =t , t =0, (ifspatiallyprojected: semicubicalparabolas y= √2k x3/2)
1 2 3
Theexcitationspectrumisanalytic,andsincetherearetwoequalcurvatureinvariantsonecan
usethedimensionlessenergy variablee k
e 2
S3/2−parab(e k )= 8p 2k√3e−2√3ek . (38)
ItisworthnotingthatS ,beingamonomialtimesanexponential,isquiteclosetothe
3/2 parab
Wien-typespectrum S (cid:181) −e 3e const.e .
W −
5. k > t , t =0,s 2 =k 2 t 2 (ifspatiallyprojected: catenaries x=k cosh(y/t ))
1 2 1
| | | | −
In general, the catenary spectrum cannot be found analytically. It is an intermediate case,
whichfort /s 0 tendstoS , whereas fort /s ¥ tendstowardS
hyp 3/2 parab
→ → −
Shyp(e k )0←t /s Scatenary(e s )t /s →¥ S3/2 parab(e k ). (39)
←− −→ −
6. t =0 (rotatingworldlinesuniformlyaccelerated normaltotheirplaneofrotation)
2
6
8
The excitation spectrum is given in this case by a two-parameter set of curves. These
trajectories are a superposition of the constant linearly accelerated motion and uniform
circular motion. The corresponding vacuum spectra have not been calculated by Letaw, not
evennumerically.
Thus,onlythehyperbolicworldlines,havingjustonenonzerocurvatureinvariant,allowfora
Planckianexcitationspetrumandleadtoastrictlyone-to-onemappingbetweenthecurvature
invariant k and the ‘thermodynamic’ temperature (T = k /2p ). The excitation spectrum
U
due to semicubical parabolas can be fitted by Wien-type spectra, the radiometric parameter
corresponding to both curvatureand torsion. The otherstationary cases, being nonanalytical,
lead to approximate determination of the curvature invariants, defining locally the classical
worldlineonwhich arelativisticquantumparticlemoves.
One can easily infer from these conclusions the reason why the Unruh effect became so
prominentwith regardtotheotherfivetypesofstationaryFrenet-Serret scalar spectra.
ImentionthatLetawintroducedtheterminologyultratorsional,paratorsional,infratorsional,
and hypertorsionalforthescalarstationarycases 3-6.
1.2. The Electromagnetic Vacuum Noise
For the case of homogeneous electromagnetic field, Honig et al proved that the FS scalars
remain constant along the worldline of a charged particle, whereas the FS vectors obey a
Lorentzforceequationoftheform
u˙m =F¯nm un (40)
withFmn =l F¯nm ,theelectromagneticfieldtensor,l =e/mc2,andu˙m thefour-velocityofthe
particle.
A remarkable physical interpretation of the FS invariants in terms of the Lorentz
invariantsoftheelectromagneticfield was establishedbyHoniget al
k 2 t 2 t 2 =l 2(E2 H2) , kt = l 2(~E H~ ) (41)
1 2 2
− − − − ·
These beautiful results passed quite unnoticed until now. They could be used for a
geometric transcription of homogeneous electromagnetism and therefore for the geometric
‘calibration’ofelectromagneticphenomena.
The FS formalism has not been used for the electromagnetic vacuum noise. Other
approaches have been undertaken for this important case of which a rather complete one
isdueto Hacyanand Sarmientothat isbriefly presented inthefollowing.
TheHacyan-Sarmientoapproach
Startingwiththeexpressionfortheelectromagneticenergy-momentumtensor
1
Tmn = 16p 4Fma Fna +h mn Flb Flb , (42)
(cid:16) (cid:17)
Hacyan–Sarmientodefinetheelectromagnetictwo-pointWightmanfunctionsas follows
1
D+mn (x,x′) 4Fa (m (x)Fn )a (x′)+h mn Flb (x)Flb (x′) ; (43)
≡ 4
(cid:16) (cid:17)
D (x,x) D+ (x,x). (44)
−mn ′ mn ′
≡
9
This may be viewed as a variant of the “point-splitting” approach advocated by DeWitt.
Moreover,becauseoftheproperties
h mn D±mn =0, D±mn =D±nm , ¶ n D±m n =0, (45)
the electromagnetic Wightman functions can be expressed in terms of the scalar Wightman
functionsas follows
D±mn (x,x′)=c¶ m ¶ n D±(x,x′), (46)
wherecisingeneralareal constantdependingonthecaseunderstudy. Thisshowsthatfrom
thestandpointoftheirvacuumfluctuationsthescalarandtheelectromagneticfieldsarenotso
different.
Nowintroducesumand differencevariables
t+t t t
s= ′; s = − ′. (47)
2 2
UsingtheFouriertransformsoftheWightmanfunctions
¥
D˜ (w ,s)= ds e iws D (s,s ), (48)
± − ±
¥
Z−
wherew isthefrequencyofzero-pointfields,theparticlenumberdensityofthevacuumseen
bythemovingdetectorand thespectralvacuum energydensityper modearegivenby
1
n(w ,s)= D˜+(w ,s) D˜ (w ,s) , (49)
(2p )2w − −
(cid:2) (cid:3)
de w 2
= D˜+(w ,s)+D˜ (w ,s) . (50)
dw p −
(cid:2) (cid:3)
Circular worldline
The most important application of these results is to a uniformly rotating detector whose
propertimeiss and angularspeed is w in motionalongthecircularworldline
0
xa (s)=(g s,R cos(w s),R sin(w s),0), (51)
0 0 0 0
where R is the rotation radius in the inertial frame, g = (1 v2) 1/2, and v =
0 −
w R /g . In this case there are two Killing vectors ka = (1,0−,0,0) and ma (s) =
0 0
(0, R sin(w s),R cos(w s,0). Expressing the Wightman functions in terms of these two
0 0 0 0
−
Killing vectors, HS calculated the following physically observable spectral quantities (i.e.,
thoseobtainedafter subtractingtheinertialzero-pointfield contributions):
Thespectralenergydensity
•
de g 3 w 2+(g vw )2 v3w2
0
= hg (w), (52)
dw 2p 3R3 w 2 w2+(2g v)2
0
Thespectralfluxdensity
•
dp g 3 w 2+(g vw )2
= 0 4v4 kg (w), (53)
dw 2p 3R3 w 2
0
Thespectralstressdensity
•
ds g 3 w 2+(g vw )2 v3w2
0
= jg (w). (54)
dw 2p 3R3 w 2 w2+(2g v)2
0
10
Theratio(w 2+(g vw )2)/w 2 isadensity-of-statesfactorintroducedforconvenience.
0
TheHacyan-Sarmientovariablesare
2w sw
0
w= x=
w 2
0
hg (w),kg (w),and jg (w)are thefollowingcosine-Fouriertransforms
¥ N (x,v) 3 2g 2v2
h
hg (w) + cos(wx)dx; (55)
≡ 0 g 2[x2 v2sin2x]3 −x4 x2
Z (cid:18) − (cid:19)
¥ N (x,v) 3 g 2
k
kg (w) cos(wx)dx; (56)
≡− 0 g 2[x2 v2sin2x]3 −x4 −6x2
Z (cid:18) − (cid:19)
¥ 1 1 2g 2v2
jg (w) + cos(wx)dx. (57)
≡ 0 g 4[x2 v2sin2x]2 −x4 3x2
Z (cid:18) − (cid:19)
ThenumeratorsN (x,v) and N (x,v) are givenby
h k
N (x,v)=(3+v2)x2+(v2+3v4)sin2x 8v2xsinx; (58)
h
−
N (x,v) =x2+v2sin2x (1+v2)xsinx. (59)
k
−
Ultra-relativisticlimit: g 1
≫
The functions Hg = v3w2 hg (w), Kg = 4v4 kg (w), Jg = v3w2 jg (w), all have
w2+(2gv)2 w2+(2gv)2
the following scaling property Xkg (kw) = k3Xg (w), where k is an arbitrary constant, and
X =H,K,J,respectively. ThisisthesamescalingpropertyasthatofaPlanckiandistribution
withatemperatureproportionalto g .
Nonrelativisticlimit: g 1
≪
AdetaileddiscussionofthenonrelativisticlimitforthescalarcasehasbeenprovidedbyKim,
Soh, and Yee [8], who used the parameters v and w , and not acceleration and speed as used
0
by Letaw and Pfautsch for the circular scalar case [9]. They obtained a series expansion in
velocity
de = w 3 w 0 (cid:229)¥ v2n (cid:229) n ( 1)k (n−k−gww 0)2n+1 Q n k w ,(60)
dw p 2 gw 2n+1 − k!(2n k)! − −gw
" n=0 k=0 − (cid:18) 0(cid:19)#
where Q is the usual Heavyside step function. Thus, to a specified power of the velocity
many vacuum harmonics could contribute; making the energy density spectrum quasi-
continuous. For low frequencies the difference between the scalar and electromagnetic case
is small. Besides, one can consider only the first few terms in the series as an already good
approximation.