Table Of ContentA WKB–like approach to Unruh Radiation
Andrea de Gill∗
Physics Department, California State University Fresno, Fresno, California 93740-8031
Douglas Singleton†
Physics Department, CSU Fresno, Fresno, CA 93740-8031
and
Institute of Gravitation and Cosmology,
Peoples’ Friendship University of Russia,
Moscow 117198, Russia
Valeria Akhmedova‡
ITEP, B. Cheremushkinskaya, 25, Moscow, Russia 117218
Terry Pilling§
0
1 Department of Physics, North Dakota State University, Fargo, North Dakota 58105-5566
0 (Dated: January 27, 2010)
2
Unruh radiation is the thermal flux seen by an accelerated observer moving through Minkowski
n spacetime. In this article we study Unruh radiation as tunneling through a barrier. We use a
a WKB–like method to obtain the tunneling rate and the temperature of the Unruh radiation. This
J derivation brings together many topics into a single problem – classical mechanics, relativity, rel-
7 ativistic field theory, quantum mechanics, thermodynamics and mathematical physics. Moreover,
2 thisgravitationalWKBmethodhelpstohighlightthefollowingsubtlepoints: (i)thetunnelingrate
strictlyshouldbewrittenastheclosedpathintegralofthecanonicalmomentum;(ii)forthecaseof
] thegravitationalWKBproblem,thereisatime–likecontributiontothetunnelingratearisingfrom
c
an imaginarychangeof the time coordinate upon crossingthe horizon. This temporal contribution
q
to the tunneling rate has no analog in the ordinary quantum mechanical WKB calculation.
-
r
g
[ I. INTRODUCTION (e.g. Reissner–Nordstrom6,deSitter8–11,KerrandKerr–
1 Newmann12,13, Unruh14). Additionally, one could easily
v incorporatetunnelingparticleswithdifferentspins15 and
3 Theradiationthatarisesfromplacingaquantumfield onecould(inasimplifiedway)begintotakeintoaccount
3 in a background metric with a horizon is a well known back reaction effects on the metric6,7,16.
8 phenomenon at the boundary between field theory and
4 general relativity. The first example of this effect was Inreference17,Unruhradiationisderivedusingpurely
1. Hawking radiation1, where a Schwarzschild black hole quantum mechanical arguments. However, the reader
needs to know the quantized radiation field, and the
0 radiates with a thermal spectrum at the expense of the
0 blackhole’smass. AnotherexampleisHawking–Gibbons mathematicalstepsinthederivationaremoreinvolvedas
1 radiation2,i.e.,thethermalradiationseenbyanobserver comparedtotheapproachpresentedhere. Incomparison
: withreference17, thegravitationalWKB–likemethodis
v in de Sitter spacetime. In this paper we focus on Unruh
i radiation3 – the radiation seen by an observer moving mathematicallysimplewhileatthesametimeitprovides
X a clear physical picture for the origins of the radiation.
with a constant acceleration through vacuum. The orig-
r In this article, this WKB–like method is presented in a
a inal methods used to calculate these effects used quan-
pedagogicalmannerforthecaseoftheRindlerspacetime
tum field theory at a level which is beyond most under-
(the metric seen by an observer who undergoes constant
graduates or beginning graduate students. In reference
proper acceleration) and Unruh radiation. The reason
1, Hawking gave a heuristic picture for the radiation in
for choosing Rindler spacetime is that it is the simplest
terms of “tunneling” of virtual particles across the hori-
spacetime in which this effect occurs. Furthermore, be-
zon. After a span of twenty five years, mathematical
detailswereaddedtothispicture4–7. Intheseworks,the cause of the strong equivalence principle (i.e., locally, a
constant acceleration and a gravitational field are obser-
action for a particle which crosses the horizon of some
vationally equivalent), the Unruh radiation from Rindler
spacetime(e.g., theSchwarzschildspacetimeforthecase
spacetime is the prototype of this type of effect. Also, of
of Hawking radiation) was calculated and found to have
all these effects – Hawking radiation, Hawking–Gibbons
an imaginary part coming from a contour integration.
radiation – Unruh radiation has the best prospects for
The exponential of this imaginary piece was compared
being observed experimentally18–21.
toaBoltzmanndistribution,whichallowedonetodeter-
mine the temperature of the radiation. The simplicity of This derivation of Unruh radiation draws together
this gravitational WKB method makes it easy to calcu- many different areas of study: (i) classical mechanics via
late Hawking like radiation for the case of other metrics theHamilton–Jacobiequations;(ii)relativityviatheuse
2
oftheRindlermetric;(iii)relativisticfieldtheorythrough Minkowski coordinates x and t, result in the familiar hy-
the Klein–Gordon equation in curved backgrounds; (iv) perbolic trajectories (i.e., x2−t2 = a−2) that represent
quantummechanicsviatheuseoftheWKB–likemethod the worldlines of the Rindler observer.
appliedtogravitationalbackgrounds; (v)thermodynam- Differentiating each coordinate in (2) and substituting
ics via the use of the Boltzmann distribution to ex- the result into (1) yields the standard Rindler metric
tractthetemperatureoftheradiation;(vi)mathematical
methodsinphysicsviatheuseofcontourintegrat ionsto ds2 =−(1+axR)2dt2R+dx2R . (3)
evaluate the imaginary part of the action of the particle
When x =−1, the determinant of the metric given by
thatcrossesthehorizon. Thusthissingleproblemserves R a
(3),det(g )≡g =−(1+ax )2,vanishes. Thisindicates
to show students how the different areas of physics are ab R
the presence of a coordinate singularity at x = −1,
interconnected. R a
which can not be a real singularity since (3) is the result
Also,throughthisexamplewewillhighlightsomesub-
of a global coordinate transformation from Minkowski
tle features of the Rindler metric and the WKB method
spacetime. ThehorizonoftheRindlerspacetimeisgiven
which are usually overlooked. In particular, we show
by x =−1.
thatthegravitationalWKBamplitudehasacontribution R a
coming from a change of the time coordinate from cross-
ing the horizon11. This temporal contribution is never
t
encounteredinordinaryquantummechanics,wheretime
acts as a parameter rather than a coordinate.
Future
II. RINDLER SPACETIMES
In this section we introduce and discuss some relevant
features of Rindler spacetime – the spacetime seen by Left wedge Right wedge
an observer moving with constant proper acceleration
through Minkowski spacetime. The Rindler metric can
x
be obtained by starting with the Minkowski metric, i.e., 1
ds2 =−dt2+dx2+dy2+dz2, where we have set c=1, 𝑎
and transforming to the coordinates of the accelerating
observer. We take the acceleration to be along the x–
direction, thus we only need to consider a 1+1 dimen-
sional Minkowski spacetime
Past
ds2 =−dt2+dx2 . (1)
Using the Lorentz transformations (LT) of special rela-
tivity, the worldlines of an accelerated observer moving
along the x–axis in empty spacetime can be related to
Minkowski coordinates t, x according to the following FIG. 1: Trajectory of the Rindler observer as seen by the
observer at rest.
transformations
t=(a−1+xR)sinh(atR) Inthespacetimediagramshownabove,thehorizonfor
(2)
x=(a−1+x )cosh(at ) , thismetricisrepresentedbythenullasymptotes,x=±t,
R R
that the hyperbola given by (2) approaches as x and t
where a is the constant, proper acceleration of the tend to infinity25. Note that this horizon is a particle
Rindler observer measured in his instantaneous rest horizon, since the Rindler observer is not influenced by
frame. One can show that the acceleration associated the whole spacetime, and the horizon’s location is ob-
with the trajectory of (2) is constant since a aµ = server dependent26.
µ
(d2x /dt2)2 = a2 with x = 0. The trajectory of (2) One can also see that the transformations (2) that
µ R R
canbeobtainedusingthedefinitionsoffour–velocityand lead to the Rindler metric in (3) only cover a quarter
four–acceleration of the accelerated observer in his in- of the full Minkowski spacetime, given by x−t >0 and
stantaneous inertial rest frame22. Another derivation of x+t > 0. This portion of Minkowski is usually labeled
(2) uses a LT to relate the proper acceleration of the Right wedge. To recover the Left wedge, one can mod-
non–inertial observer to the acceleration of the inertial ify the second equation of (2) with a minus sign in front
observer23. The text by Taylor and Wheeler24 also pro- of the transformation of the x coordinate, thus recover-
vides a discussion of the Rindler observer. ing the trajectory of an observer moving with a negative
The coordinates x and t , when parametrized and acceleration. In fact, we will show below that the coor-
R R
plotted in a spacetime diagram whose axes are the dinatesx andt doublecovertheregioninfrontofthe
R R
3
horizon, x = −1. In this sense, the metric in (3) is By applying the WKB method to this scalar field, we
R a
similar to the Schwarzschild metric written in isotropic findthatthephaseofthescalarfielddevelopsimaginary
coordinates. For further details, see reference 26. contributions upon crossing the horizon. The exponen-
ThereisanalternativeformoftheRindlermetricthat tial of these imaginary contributions is interpreted as a
canbeobtainedfrom(3)bythefollowingtransformation: tunnelingamplitudethroughthehorizon. Byassuminga
Boltzmann distribution and associating it with the tun-
(cid:112)
(1+ax )= |1+2ax | . (4) nelingamplitude, weobtainthetemperatureoftheradi-
R R(cid:48)
ation.
Using the coordinate transformation given by (4) in Writing the scalar field in terms of a phase factor as
(3), we get the following Schwarzschild–like form of the φ = φ0e(cid:126)iS(t,(cid:126)x), the Hamilton–Jacobi equations for the
Rindler metric
action S of the field φ in the gravitational background
given by the metric g are (see Appendix I for details)
ds2 =−(1+2ax )dt2 +(1+2ax )−1dx2 . (5) µν
R(cid:48) R(cid:48) R(cid:48) R(cid:48)
gµν∂ (S)∂ (S)+m2 =0 . (8)
If one makes the substitution a → GM/x2 one can see ν µ
R(cid:48)
the similarity to the usual Schwarzschild metric. The Now for stationary spacetimes (technically spacetimes
horizon is now at x =−1/2a and the time coordinate, for which one can define a time–like Killing vector that
R(cid:48)
t , doeschangesignasonecrossesx =−1/2a. Inad- yields a conserved energy, E) the action S can be split
R(cid:48) R(cid:48)
dition, from (4) one can see explicitly that as x ranges into a time and space part, i.e., S(t,(cid:126)x)=Et+S ((cid:126)x).
R(cid:48) 0
from+∞to−∞thestandardRindlercoordinatewillgo If S has an imaginary part, this then gives the tun-
0
from+∞downtox =−1/aandthenbackoutto+∞. neling rate, Γ , via the standard WKB formula. The
R QM
The Schwarzschild–like form of the Rindler metric WKB approximation tells us how to find the transmis-
given by (5) can also be obtained directly from the 2– sion probability in terms of the incident wave and trans-
dimensional Minkowski metric (1) via the transforma- mitted wave amplitudes. The transition probability is in
tions turngivenbytheexponentiallydecayingpartofthewave
√ function over the non–classical (tunneling) region29
1+2ax
t= R(cid:48) sinh(at )
√ a R(cid:48) (6) ΓQM ∝e−Im(cid:126)1(cid:72) pxdx . (9)
1+2ax
x= a R(cid:48) cosh(atR(cid:48)) The tunneling rate given by (9) is just the lowest order,
quasiclassicalapproximationtothefullnon–perturbative
for x ≥− 1 , and Schwinger30 rate.37
R(cid:48) 2a
In most cases (with an important exception that we
(cid:112)|1+2ax | will discuss in appendix II), pout and pin have the same
R(cid:48)
t= a cosh(atR(cid:48)) magnitude but opposite signs. Thus ΓQM will receive
(cid:112) (7) equal contributions from the ingoing and outgoing parti-
|1+2ax |
x= R(cid:48) sinh(at ) cles,sincethesigndifferencebetweenpout andpin willbe
a R(cid:48) compensated for by the minus sign that is picked up in
the pin integration due to the fact that the path is being
for x ≤− 1 . Note that imposing the above conditions
R(cid:48) 2a traversed in the backward x-direction. In all quantum
on the coordinate x fixes the signature of the metric,
R(cid:48) mechanical tunneling problems that we are aware of this
since for x ≤ − 1 or 1+2ax ≤ 0 the metric signa-
R(cid:48) 2a R(cid:48) isthecase: thetunnelingrateacrossabarrieristhesame
turechangesto(+,−),whilefor1+2ax ≥0themetric
R(cid:48) for particles going right to left or left to right. For this
has signature (−,+). Thus one sees that the crossing of
reason, the tunneling rate (9) is usually written as29
the horizon is achieved by the crossing of the coordinate
scianugsuelsartihtey,rwadhiiacthioins pinrecthisieslyfotrhmeatliusnmn.elAinsgabafirnriaelrctohmat- ΓQM ∝e∓2Im(cid:126)1(cid:82) poxut,indx , (10)
ment, we note that the determinant of the metric for (3) In (10) the − sign goes with pout and the + sign with
x
is zero at the horizon xR =−1/a, while the determinant pixn.
of the metric given by (5) is 1 everywhere. There is a technical reason to prefer (9) over (10).
As was remarked in references 32–34, equation (9) is
invariant under canonical transformations, whereas the
III. THE WKB/TUNNELING METHOD form given by (10) is not. Thus the form given by (10)
is not a proper observable. Moreover, in appendix II,
Inthissectionwestudyascalarfieldplacedinaback- we will show an example of the WKB method for the
ground metric. Physically, these fields come from the Schwarzschild spacetime in Painlev´e-Gulstrand coordi-
quantum fields, i.e., vacuum fluctuations, that permeate nates, and we will find that the two formulas, (9) and
thespacetimegivenbythemetric. Inaddition,asshown (10), are not numerically equivalent.
in reference 27, the vacuum field fluctuations obey the However, for the case of the gravitational WKB prob-
principle of equivalence, which supports this approach. lem, equation (10) only gives the imaginary contribution
4
to the total action coming from the spatial part of the From (15), S is found to be
0
action. In addition, there is a temporal piece, E∆t, that (cid:90) ∞ (cid:112)E2−m2(1+2ax )
mustbeaddedtothetotalimaginarypartoftheactionto S± =± R(cid:48) dx . (16)
obtainthetunnelingrate. Thistemporalpieceoriginates 0 −∞ 1+2axR(cid:48) R(cid:48)
from an imaginary change of the time coordinate as the
In (16), the + sign corresponds to the ingoing particles
horizoniscrossed. Wewillexplicitlyshowhowtoaccount
(i.e.,particlesthatmovefromrighttoleft)andthe−sign
for this temporal piece in the next section, where we ap-
to the outgoing particles (i.e., particles that move left to
ply the WKB method to the Rindler spacetime. This (cid:82)
right). Note also that (16) is of the form S = p dx,
0 x
imaginary part of the total action coming from the time
where p is the canonical momentum of the field in the
x
pieceis auniquefeatureofthe gravitationalWKBprob-
Rindler background. The Minkowski spacetime expres-
lem. Therefore, for the case of the gravitational WKB
sion for the momentum is easily recovered by setting
√
problem, the tunneling rate is given by
a=0, in which case one sees that p = E2−m2.
x
From (16), one can see that this integral has a pole
Γ∝e−(cid:126)1[Im((cid:72) pxdx)−EIm(∆t)] . (11) along the path of integration at x = − 1 . Using a
R(cid:48) 2a
contour integration gives an imaginary contribution to
In order to obtain the temperature of the radiation, we the action. We will give explicit details of the contour
assume a Boltzmann distribution for the emitted parti- integration since this will be important when we try to
cles apply this method to the standard form of the Rindler
metric (3) (see Appendix III for the details of this cal-
Γ∝e−ET , (12) culation). We go around the pole at x = − 1 us-
R(cid:48) 2a
ing a semi–circular contour which we parameterize as
where E is the energy of the emitted particle, T is the x = − 1 +(cid:15)eiθ, where (cid:15) (cid:28) 1 and θ goes from 0 to
temperature associated with the radiation, and we have R(cid:48) 2a
π for the ingoing path and π to 0 for the outgoing path.
set Boltzmann’s constant, k , equal to 1. Eq. (12) gives
B These contours are illustrated in the figure below. With
the probability that a system at temperature T occupies
this parameterization of the path, and taking the limit
a quantum state with energy E. One weak point of this
(cid:15)→0,wefindthattheimaginarypartof (16)foringoing
derivation is that we had to assume a Boltzmann distri-
(+) particles is
bution for the radiation while the original derivations 1,3 √
obtain the thermal spectrum without any assumptions. (cid:90) π E2−m2(cid:15)eiθ iπE
S+ = i(cid:15)eiθdθ = , (17)
Recently, this shortcoming of the tunneling method has 0 2a(cid:15)eiθ 2a
0
been addressed in reference 35, where the thermal spe c-
and for outgoing (−) particles, we get
trum was obtained within the tunneling method using
density matrix techniques of quantum mechanics. √
By equating (12) and (11), we obtain the following (cid:90) 0 E2−m2(cid:15)eiθ iπE
S− =− i(cid:15)eiθdθ = . (18)
formula for the temperature T 0 2a(cid:15)eiθ 2a
π
E(cid:126)
T = . (13)
Im(cid:0)(cid:72) p dx(cid:1)−EIm(∆t) (i)
x
IV. UNRUH RADIATION VIA (ii)
WKB/TUNNELING
We now apply the above method to the alternati ve FIG. 2: Contours of integration for (i) the ingoing and (ii)
Rindler metric previously introduced. For the 1 + 1 the outgoing particles.
Rindler spacetimes, the Hamilton–Jacobi equations (H–
J) reduce to gtt∂ S∂ S+gxx∂ S∂ S+m2 = 0. For the
t t x x InordertorecovertheUnruhtemperature,weneedto
Schwarzschild–like form of Rindler given in (5) the H–J
take into account the contribution from the time piece
equations are
of the total action S(t,(cid:126)x) = Et+S ((cid:126)x), as indicated by
0
the formula of the temperature, (13), found in the previ-
1
− (∂ S)2+(1+2ax )(∂ S)2+m2 =0 . ous section. The transformation of (6) into (7) indicates
(1+2ax ) t R(cid:48) x
R(cid:48) that the time coordinate has a discrete imaginary jump
(14)
as one crosses the horizon at x = − 1 , since the two
Now splitting up the action S as S(t,(cid:126)x)=Et+S ((cid:126)x) R(cid:48) 2a
0 timecoordinatetransformationsareconnectedacrossthe
in (14) gives
horizon by the change t →t − iπ, that is,
R(cid:48) R(cid:48) 2a
E m2 (cid:18) iπ(cid:19)
− +(∂ S (x ))2+ =0. (15) sinh(at )→sinh at − =−icosh(at ) .
(1+2ax )2 x 0 R(cid:48) 1+2ax R(cid:48) R(cid:48) 2 R(cid:48)
R(cid:48) R(cid:48)
5
Note that as the horizon is crossed, a factor of i comes the same parts of the full spacetime diagram. Also, as
from the term in front of the hyperbolic function in (6), one crosses the horizon, there is an imaginary jump of
i.e., the Rindler time coordinate as given by comparing (6)
√ and (7).
(cid:112)
1+2axR(cid:48) →i |1+2axR(cid:48)| , Inaddition,forthegravitationalWKBproblem,Γhas
contributions from both the spatial and temporal parts
so that (7) is recovered.
of the action. Both these features are not found in the
Therefore every time the horizon is crossed, the to-
ordinary quantum mechanical WKB problem.
tal action S(t,(cid:126)x) = S ((cid:126)x) + Et picks up a factor of
0 As a final comment, note that one can define an ab-
E∆t = −iπE. For the temporal contribution, the direc-
tion in wh2icah the horizon is crossed does not affect the sorption probability (i.e., Pabs ∝|φin|2) and an emission
probability (i.e., P ∝ |φ |2). These probabilities
sign. This is different from the situation for the spatial emit out
can also be used to obtain the temperature of the radia-
contribution. Whenthehorizoniscrossedonce,thetotal
tion via the “detailed balance method”5
action S(t,(cid:126)x) gets a contribution of E∆t = −iEπ, and
for a round trip, as implied by the spatial part 2(cid:72)a pxdx, Pemit =e−E/T .
the total contribution is E∆t = −iEπ. So using the P
total a abs
equation for the temperature (13) developed in the pre-
vious section, we obtain Using the expression of the field φ = φ0e(cid:126)iS(t,(cid:126)x), the
Schwarzschild–like form of the Rindler metric given in
E(cid:126) (cid:126)a
(5), and taking into account the spatial and temporal
T = = , (19)
Unruh πE + πE 2π contributions gives an an absorption probability of
a a
whichistheUnruhtemperature. Theinterestingfeature Pabs ∝eπaE−πaE =1
of this result is that the gravitational WKB problem has
contributionsfrombothspatialandtemporalpartsofthe
and an emission probability of
wave function, whereas the ordinary quantum mechani-
cal WKB problem has only a spatial contribution. This Pemit ∝e−πaE−πaE =e−2πaE .
is natural since time in quantum mechanics is treated as
adistinctparameter, separateincharacterfromthespa- The first term in the exponents of the above probabili-
tial coordinates. However, in relativity time is on equal ties corresponds to the spatial contribution of the action
footing with the spatial coordinates. S, while the second term is the time piece. When using
thismethod,wearenotdealingwithadirectedlineinte-
gral as in (9), so the spatial parts of the absorption and
V. CONCLUSION
emission probability have opposite signs. In addition,
the absorption probability is 1, which physically makes
WehavegivenaderivationofUnruhradiationinterms sense – particles should be able to fall into the horizon
of the original heuristic explanation as tunneling of vir- with unit probability. If the time part were not included
tual particles tunneling through the horizon1. This tun- in P , then for some given E and a one would have
abs
neling method can easily be applied to different space- Pabs ∝eπaE >1, i.e., the probability of absorption would
timesandtodifferenttypesofvirtualparticles. Wechose exceed 1 for some energy. Thus for the detailed balance
the Rindler metric and Unruh radiation since, because method the temporal piece is crucial to ensure that one
of the local equivalence of acceleration and gravitational has a physically reasonable absorption probability.
fields, it represents the prototype of all similar effects
(e.g. Hawking radiation, Hawking–Gibbons radiation). Acknowledgments
Since this derivation touches on many different ar-
eas – classical mechanics (through the H–J equations), TheauthorswouldliketothankE.T.Akhmedovforvalu-
relativity (via the Rindler metric), relativistic field the- able discussions. D.S. is supported by a 2008-2009 Ful-
ory(throughtheKlein–Gordonequationincurvedback- bright Scholars Grant. D.S. would like to thank Vitaly
grounds), quantum mechanics (via the WKB method Melnikov for the invitation to research at the Center for
for gravitational fields), thermodynamics (via the Boltz- Gravitation and Fundamental Metrology and the Insti-
mann distribution to extract the temperature), and tute of Gravitation and Cosmology at PFUR. The au-
mathematical methods (via the contour integration to thors would also like to thank two anonymous referees
obtain the imaginary part of the action) – this single from this journal for their valuable comments and sug-
problemservesasareminderoftheconnectionsbetween gestions that led to the final version of this paper.
the different areas of physics.
This derivation also highlights several subtle points Appendix I: The Hamilton–Jacobi equations
regarding the Rindler metric and the WKB tunneling
method. In terms of the Rindler metric, we found that TheHamilton–Jacobiequationsmaybederivedfromthe
thedifferentformsofthemetric(3)and(5)donotcover Klein–Gordon equation in the following manner. The
6
Klein–Gordon (KG) equation for a scalar field φ of mass Taking the classical limit, i.e., letting (cid:126) → 0, we obtain
m, placed in a background metric g is the Hamilton–Jacobi equations for the action S of the
µν
fieldφinthegravitationalbackgroundgivenbythemet-
(cid:18) 1 √ m2c2(cid:19) ric g ,
√ ∂ ( −ggµν∂ )− φ=0 , (20) µν
−g µ ν (cid:126)2
gµν∂ (S)∂ (S)+m2 =0 . (28)
ν µ
where c is the speed of light and (cid:126) is Planck’s con-
stant. For Minkowski spacetime, the above reduces to Appendix II: Hawking radiation from the
the free Klein–Gordon equation, i.e., ((cid:3)−m2c2/(cid:126)2)φ = Painlev´e–Gulstrand form of the Schwarzschild
(−∂2/c2∂t2+∇2−m2c2/(cid:126)2)φ=0. spacetime
In using a scalar field, we are following the original
ThePainlev´e–GulstrandformoftheSchwarzschildspace-
works1,3. Despite the fact that, absent the hypotheti-
time is obtained by transforming the Schwarzschild time
cal Higgs boson, there are no known fundamental scalar
t to the Painlev´e–Gulstrand time t(cid:48)
fields,thederivationwithspinororvectorparticleswould
only add the complication of having to carry around (cid:113)
2M dr
spinor or Lorentz indices without adding to the basic dt=dt(cid:48)− r . (29)
understanding of the phenomenon. Using the WKB ap- 1− 2M
r
proach presented here it is straightforward to do the cal-
culation using spinor15 or vector particles. Applying the above transformation to the Schwarzshild
Settingthespeedoflightcequalto1,multiplying(20) metric gives the Painlev´e–Gulstrand form of the
by −(cid:126) and using the product rule, (20) becomes Schwarzschild spacetime
−(cid:126)2 (cid:104) √ √ (cid:18) 2M(cid:19) (cid:114)2M
√ (∂ −g)gµν∂ φ+ −g(∂ gµν)∂ φ+ ds2 =− 1− dt(cid:48)2+2 drdt(cid:48)+dr2 . (30)
−g µ ν µ ν r r
(21)
√ (cid:105)
−ggµν∂µ∂νφ +m2φ=0 . The time is transformed, but all the other coordinates
(r,θ,φ) are the same as the Schwarzschild coordinates.
The above equation can be simplified using the fact that If we use the metric in (30) to calculate the spatial part
the covariant derivative of any metric g vanishes of the action as in (35) and (16), we obtain
∇αgµν =∂αgµν +Γµαβgβν +Γναβgµβ =0 , (22) S = −(cid:90) ∞ dr (cid:114)2M E (31)
0 1− 2M r
where Γµ is the Christoffel connection. All the metrics −∞ r
αβ (cid:115)
that we consider here are diagonal so Γµ = 0, for µ (cid:54)= (cid:90) ∞ dr (cid:18) 2M(cid:19)
αβ ± E2−m2 1− . (32)
α(cid:54)=β. It can also be shown that 1− 2M r
−∞ r
√
Γµ =∂ (ln√−g)= ∂√γ −g . (23) Each of these two integrals has an imaginary contribu-
µγ γ −g tion of equal magnitude, as can be seen by performing a
contour integration. Thus one finds that for the ingoing
Using(22)and(23),theterm∂µgµν in(21)canberewrit- particle(the+signinthesecondintegral)onehasazero
ten as net imaginary contribution, while from for the outgoing
√ particle(the−signinthesecondintegral)thereisanon–
∂ −g
∂ gµν =−Γµ gγν −Γν gµρ =− √γ gγν , (24) zero net imaginary contribution. Therefore in this case
µ µγ µρ −g
there is clearly a difference by a factor of two between
using (9) and (10) which comes exactly because the tun-
since the harmonic condition is imposed on the metric
neling rates from the spatial contributions in this case
gµν, i.e., Γν gµρ =0. Thus (21) becomes
µρ do depend upon the direction in which the barrier (i.e.,
−(cid:126)2gµν∂ ∂ φ+m2φ=0 . (25) the horizon) is crossed. The Schwarzcshild metric has a
µ ν similartemporalcontributionasfortheRindlermetric36.
We now express the scalar field φ in terms of its action The Painlev´e–Gulstrand form of the Schwarzschild met-
S =S(t,(cid:126)x) ricactuallyhastwotemporalcontributions: (i)onecom-
ing from the jump in the Schwarzschild time coordinate
φ=φ0e(cid:126)iS(t,(cid:126)x) , (26) similar to what occurs with the Rindler metric in (6)
and (7); (ii) the second temporal contribution coming
where φ is an amplitude28 not relevant for calculating from the transformation between the Schwarzschild and
0
the tunneling rate. Plugging this expression for φ into Painlev´e–Gulstrand time coordinates in (29). If one in-
(25), we get tegrates equation (29), one can see that there is a pole
coming from the second term. One needs to take into
−(cid:126)gµν(∂ (∂ (iS)))+gµν∂ (S)∂ (S)+m2 =0 . (27) account both of these time contributions in addition to
µ ν ν µ
7
the spatial contribution, to recover the Hawking tem- The above yields the following solution for S
0
perature. Only by adding the temporal contribution to
the spatial part from (9), does one recover the Hawking
temperature36 T = (cid:126) . Thusforbothreasons–canon-
8πM
ical invariance and to recover the temperature – it is (9)
which should be used over (10), when calculating ΓQM. S± =±(cid:90) ∞ (cid:112)E2−m2(1+axR)2 dx , (35)
In ordinary quantum mechanics, there is never a case – 0 1+ax R
−∞ R
as far as we know – where it makes a difference whether
one uses (9) or (10). This feature – dependence of the
tunnelingrateonthedirectioninwhichthebarrieristra-
verse–appearstobeauniquefeatureofthegravitational
WKB problem. So in terms of the WKB method as ap- where the +(−) sign corresponds to the ingoing (outgo-
plied to the gravitational field, we have found that there ing) particles.
aresituations(e.g. SchwarzschildspacetimeinPainlev´e–
Gulstrandcoordinates)wherethetunnelingratedepends
Looking at (35), we see that the pole is now at x =
R
on the direction in which barrier is traversed so that (9)
−1/a and a naive application of contour integration ap-
over (10) are not equivalent and will thus yield different pears to give the results ±iπE. However, this cannot
tunneling rates, Γ. a
be justified since the two forms of the Rindler metric
– (3) and (5) – are related by the simple coordinate
Appendix III: Unruh radiation from the
transformation (4), and one should not change the value
standard Rindler metric
of an integral by a change of variables. The resolu-
tion to this puzzle is that one needs to transform not
ForthestandardformoftheRindlermetricgivenby(3),
only the integrand but the path of integration, so apply-
the Hamilton–Jacobi equations become
ing the transformation (4) to the semi–circular contour
√
1 x =− 1 +(cid:15)eiθ givesx =−1+ (cid:15)eiθ/2. Becauseeiθ is
− (1+axR)2 (∂tS)2+(∂xS)2+m2 =0 . (33) reRp(cid:48)laced2bay eiθ/2 due toRthe sqauarearoot in the transfor-
mation (4), the semi–circular contour of (17) is replaced
After splitting up the action as S(t,(cid:126)x)=Et+S ((cid:126)x), we
0 by a quarter–circle, which then leads to a contour inte-
get gral of iπ ×Residue instead of iπ×Residue. Thus both
2
forms of Rindler yield the same spatial contribution to
E
− +(∂ S (x ))2+m2 =0 . (34) the total imaginary part of the action.
(1+ax )2 x 0 R
R
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