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Apparent correction to the speed of light in a gravitational potential

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2014 New J. Phys. 16 065008

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Apparent correction to the speed of light in agravitational potential

J D FransonPhysics Department, University of Maryland, Baltimore County, Baltimore, MD 21250, USAE-mail: [email protected]

Received 20 December 2013, revised 13 March 2014Accepted for publication 10 April 2014Published 12 June 2014

New Journal of Physics 16 (2014) 065008

doi:10.1088/1367-2630/16/6/065008

AbstractThe effects of physical interactions are usually incorporated into the quantumtheory by including the corresponding terms in the Hamiltonian. Here weconsider the effects of including the gravitational potential energy of massiveparticles in the Hamiltonian of quantum electrodynamics. This results in apredicted correction to the speed of light that is proportional to the ne structureconstant. The correction to the speed of light obtained in this way depends on thegravitational potential and not the gravitational eld, which is not gaugeinvariant and presumably nonphysical. Nevertheless, the predicted results are inreasonable agreement with experimental observations from Supernova 1987a.

Keywords: quantum, gravity, light, potential

1. Introduction

One might suppose that the effects of a gravitational eld on a quantum system could bedescribed, at least to a rst approximation, by including the gravitational potential G in theHamiltonian. That approach has been successfully used, for example, to analyze the results ofneutron [1] and atom [210] interferometer experiments in a gravitational eld. Here weconsider a model in whichG is included in the Hamiltonian of quantum electrodynamics. As aresult, virtual electron-positron pairs [1116] have a gravitational potential energy that is thesame as that of real particles. A straightforward calculation based on that assumption shows that

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the velocity of light in a gravitational potential would be reduced by an amount that isproportional to the ne structure constant .

The predicted correction to the speed of light depends on the gravitational potential and notthe gravitational eld, and it could be observed locally by comparing the velocity of photonsand neutrinos, for example. As a result, the predicted correction to the speed of light is notgauge invariant. These results are also not equivalent to what would be obtained [1719] fromthe currently-accepted generalization of the Dirac equation and quantum electrodynamics tocurved spacetime [2025]. The lack of gauge invariance and the disagreement with thegenerally-covariant Dirac equation both suggest that including the gravitational potential in theHamiltonian must be nonphysical.

Nevertheless, the predicted correction to the speed of light from this simple model is inreasonable agreement with experimental observations from Supernova 1987a, where the rstneutrinos arrived approximately 7.7 h before the rst photons [26]. There is no conventionalexplanation for how that could have occurred and the currently-accepted interpretation of thedata is that the rst burst of neutrinos must have been unrelated to the supernova [26], despitethe fact that the probability of such an event having occurred at random is less than 10 4 [27].The predicted correction to the speed of light, if correct, could explain this long-standinganomaly.

Quantum mechanics and general relativity are two of the most fundamental laws ofphysics. Quantum mechanics has been veried to very high precision by quantumelectrodynamics experiments such as the measurement of the electron g-factor [28, 29].Experimental tests of general relativity are much more limited and many of the observedphenomena are consistent with other formalisms. As a result, there is currently a great deal ofinterest in performing high-precision tests [30] of general relativity using the properties ofquantum systems, such as atom interferometers [210] superconductors [3134], and photons[35, 36]. The correction to the speed of light predicted here is closely related to the equivalenceprinciple, as will be described below, and these results may provide additional motivation forexperimental tests of general relativity, especially the equivalence principle.

Einstein was the rst to predict that the velocity of light would be reduced by agravitational potential [37]. According to general relativity [38, 39], the speed of light c asmeasured in a global reference frame is given by

= + ( )rc c

c1 2 , (1)G0

02

where c0 is the speed of light as measured in a local freely-falling reference frame. Thisreduction in the speed of light can be observed if a beam of light passes near a massive objectsuch as the Sun, as illustrated in gure 1. The transit time from a distant planet or satellite toEarth can be measured as a function of the distance D of closest approach to the Sun and thencompared to the transit time expected at a velocity of c0. The results from such experiments [40]are in excellent agreement with the prediction of equation (1). The deection of starlight by amassive object can also be intuitively understood in this way. It should be noted that thevelocity of light measured by a local observer will be independent ofG and that the observableeffects in this example are due to the spatial variations in G.

The model considered here gives a correction to equation (1) that is proportional to the nestructure constant. These results are based on the Feynman diagrams [1116, 41] of gure 2 as

New J. Phys. 16 (2014) 065008 J D Franson

2


3

Figure 1. A measurement of the transit time at the speed of light from a distant satelliteto Earth. Einstein predicted that the speed of light as measured in a global referenceframe would be reduced by the gravitational potential of the Sun as described byequation (1), which is in good agreement with experiments. Here D is the distance ofclosest approach. The deection of the light beam by the gravitational potential of theSun is very small and is not illustrated here.

Figure 2. (a) A Feynman diagram in which a photon with wave vector k is annihilatedto produce a virtual state containing an electron with momentum p and a positron withmomentum q. After a short amount of time, the electron and positron are annihilated toproduce a photon with the original wave vector k. Any effect that this process may haveon the velocity of light is removed using renormalization techniques to give theobserved value of c0. (b) The same process, except that now the energies of the virtualelectron and positron include their gravitational potential energy m G, as indicated bythe arrows. This produces a small change in the velocity of light that is experimentallyobservable. The variable t represents the time while x represents the position in threedimensions (in arbitrary units).

will be described in more detail below. Roughly speaking, the gravitational potential changesthe energy of a virtual electron-positron pair, which in turn produces a small change ( )E k inthe energy of a photon with wave vector k as can be shown using perturbation theory. Thisresults in a small correction to the angular frequency ( )k of a photon and thus its velocity

= ( )c k k. The analogous effects for neutrinos involve the weak interaction and they arenegligibly small in comparison. As a result, this model predicts a small but observable reductionin the velocity of photons relative to that of neutrinos. In principle, the reduction in the speed oflight could be directly measured by a local observer, but a small change in c can be more easilyobserved by comparing the photon and neutrino velocities.

The remainder of the paper begins with the motivation for including the gravitationalenergy m G in the Hamiltonian of a quantum system. The correction to the speed of light due tothe gravitational potential is then calculated using quantum electrodynamics and standardperturbation theory in section 3. Gauge invariance and the equivalence principle are discussedin section 4, while the predicted delay in the photon arrival time for Supernova 1987a iscalculated in section 5 and compared with the experimental data. A summary and conclusionsare presented in section 6. Appendix A discusses the role of gravitational potentials in thetheory of general relativity and the well-known analogy with electromagnetism. Appendix Bshows that the relativistic Hamiltonian considered here correctly reduces to the Pauli equationin the nonrelativistic limit for both electrons and positrons.

2. Gravitational potentials in the quantum theory

In nonrelativistic quantum mechanics, the Hamiltonian H for a particle with mass m in aNewtonian gravitational eld is given by

= + ( )rHm

m2

(2)G2

2

where ( )rG is the Newtonian gravitational potential at position r. Equation (2) has beensuccessfully used to analyze the results of neutron interferometer experiments in a gravitationaleld, for example [1].

Equation (2) can be generalized to include the nonNewtonian gravitational effects of arotating mass M by making use of the well-known analogy [33, 4249] betweenelectromagnetism and Einsteins eld equations for a weak gravitational eld. As outlined inappendix A, it is possible to dene a gravitational vector potential ( )A r t,G and a gravitationalscalar potential ( )r t,G that are determined by the metric. The motion of a classical particle isthen described by the usual Lorentz force equation, which provides a convenient way tovisualize general relativistic effects such as frame-dragging. This suggests that equation (2) canbe generalized to

=

+( ) ( )A r rHm i

m

ct m t

12

, , . (3)G G

2

Equation (3) has previously been used in connection with the gravitational analog of theAharonovBohm effect, for example [33, 47, 5052]. We will be interested here in situations


4

where the source of the gravitational eld is stationary in the chosen coordinate frame, in whichcase =A 0G and equation (3) reduces to equation (2).

Equations (2) and (3) suggest that it may be possible to represent the effects of a weakgravitational eld in quantum electrodynamics, at least to a rst approximation, by including

m G in the usual interaction Hamiltonian H . This gives

= +

+

( ) ( ) ( ) ( )

( ) ( )

rj r A r r r r

r r r

Hc

d t t d t t

d t t

1, , , ,

, , (4)

E E E E

G G

3 3

3

in the Lorentz gauge [12, 16]. Here the electromagnetic charge density ( )r t,E and currentdensity ( )j r t,E are given as usual by

=

=

( ) ( ) ( )( ) ( ) ( )r r r

j r r r

t q t t

t cq t t

, , ,

, , , . (5)E

E

The charge of an electron is denoted by q, ( )r t, is the Dirac eld operator, represents theDirac matrices [53], and ( )r t,G corresponds to the mass density of the particles as described inmore detail in appendix B. ( )A r t,E and ( )r t,E represent the vector and scalar potentials of theelectromagnetic eld and the rst two terms in equation (4) correspond to the usual interactionbetween charged particles and the electromagnetic eld. The third term represents thegravitational potential energy of any particles. It is shown in appendix B that the interactionHamiltonian of equation (4) correctly reduces to the Pauli equation in the nonrelativistic limitwith the correct sign of m G for both electrons and positrons. This is equivalent to theSchrdinger equation of equation (2) in the absence of a magnetic eld.

For our purposes, the gravitational potentials AG and G will be assumed to be classicalelds. Similar results would be obtained if a weak gravitational eld were quantized tointroduce gravitons, as is illustrated in gure 3.

Equation (4) represents a simple model in which the gravitational potential of any massiveparticles is included in the Hamiltonian. Although this assumption seems plausible, it leads toan observable correction to the speed of light that is not equivalent to what is obtained using thecurrently-accepted generalization of the Dirac equation to curved spacetime [1719]. The factthat the predictions of this simple model are in reasonable agreement with experimentalobservation may provide some motivation for considering the differences between these twoapproaches.

3. Calculated correction to the speed of light

In quantum electrodynamics, there is a probability amplitude for a photon propagating in freespace with wave vector k and angular frequency = ck to be annihilated while producing avirtual state containing an electron-positron pair, as illustrated in gure 2(a). The virtual stateonly exists for a brief amount of time, after which the process is reversed and the electron-positron pair is annihilated and the original photon is reemitted. This process, which is knownas vacuum polarization [54], leads to divergent terms that can be eliminated using


5

renormalization techniques while small corrections to this process can produce observableeffects.

Here we will calculate the change E in the energy of a photon with wave vector k due tothe interaction of a virtual electron-positron pair with the gravitational potential as illustratedschematically in gure 2(b). The gravitational potential changes the energy of the virtualelectronpositron state by m2 G, as is shown in appendix B. That in turn changes the energy ofthe photon by a small amount as will be shown below using perturbation theory. Thedependence of the energy (and thus the frequency ) of a photon on k will produce a correctionto its velocity k. It will be assumed here that the gravitational eld is not quantized and isdescribed by the Newtonian potential G.

From second-order perturbation theory [53], the change E ( )2 in the energy of a photonwith wave vector k is given by

=

k kE

H n n H

E E. (6)( )

( ) ( )n n

2

00 0

Here k represents the unperturbed initial state containing only the photon while n representsall possible intermediate states containing an electron-positron pair. The unperturbed energy of

the initial state is represented by E ( )00 while E ( )n

0 is the unperturbed energy of the intermediatestate. For the purposes of this calculation, the gravitational potential term in equation (4) will be

included in the unperturbed Hamiltonian H0 while the electromagnetic interaction terms will be

included in the perturbation Hamiltonian H . As a result, the unperturbed energy of theintermediate state containing an electron-positron pair becomes = + +E E E m2( )n p q G0 . Here pand q are the momenta of the virtual electron and positron respectively, as in gure 2, while Ep

is the relativistic energy of a free particle given as usual by = +E p c m cp 2 2 2 4 .


6

Figure 3. Feynman diagrams that are equivalent to those of gure 2 except that heregravity is quantized and the gravitational potential is produced by the emission andabsorption of virtual gravitons. (a) A graviton produced by mass M is absorbed by avirtual electron, changing its momentum from p to p . (b) A virtual positron emits agraviton that is then absorbed by mass M . The momentum of the positron is changedfrom q to q . Two other diagrams (not shown) involve the emission of a graviton by avirtual electron or the absorption of a graviton by a virtual positron.

Straightforward perturbation theory will be used for simplicity and because the usualFeynman diagram rules [1116] may not be directly applicable to the Hamiltonian of equation(4). Equation (6) corresponds to steady state perturbation theory, but the same results can beobtained using the forward-scattering amplitude from time-dependent perturbation theory [53].

We will use periodic boundary conditions with a unit volume V [55]. In the Schrodingerpicture, the Dirac eld operators are then given by [15]

= + ( ) ( ) ( ) ( ) ( )r p p p pmcE b s u s e c s v s e, , , , . (7)p pp r p r

s

i i

,

2

Here ( )pb s, creates an electron with momentum p and spin s, whose values will be denotedby to indicate spin up or down, while ( )pc s, creates a positron with momentum p and spin s.The Dirac spinors ( )pu s, and ( )pv s, are dened [14, 15, 53] by

+ =+

+

+

=+

++

=+

+

++ =

+++

+

+

( ) ( )

( ) ( )

p p

p p

uE mc

mc

p c

E mcp c

E mc

uE mc

mc

p c

E mcp c

E mc

vE mc

mc

p c

E mcp c

E mcv

E mc

mc

p c

E mcp c

E mc

,2

10

,2

01

,2

10

,2

01

(8)

p

p

p

p

p

p

p

p

p

p

p

p

z

z

z

z

2

22

2

2

22

2

2

2

2

2

2

2

2

2

where p p ipx y.The scalar electromagnetic potential ( )rE and the longitudinal part of ( )A rE do not

contribute to this process. In Gaussian units, the transverse part of the electromagnetic vectorpotential operator is given by

= +

( )A r c a e a e2 . (9)k

kk r

kk r

Ei i

,

2

Here = ck and denotes two transverse polarization unit vectors. Without loss of generality,we can assume that the initial photon has its wave vector k in the x direction with itspolarization along the z direction.

The integral over r in the interaction Hamiltonian of equation (4) combined with theexponential factors in ( )r , ( )r , and ( )A r t,E give a delta-function that conserves momentum,so that = q k p and the sum over intermediate states reduces to a sum over all values of p.We will assume that the energy of the photon is sufciently small that kc mc2, in whichcase E Eq p (i.e., the recoil momentum from absorbing the photon has a negligible effect on the


7

virtual particle energies). For the same reason, we can approximate q by p in the evaluation of ( ) ( )p qu s v s, ,z with the result that

+ =

++

+

+ ( )

( )( ) ( )p qu v

E mc

mc

p p p c

E mc, ,

21 , (10)z

p x y z

p

2

2

2 2 2 2

2 2

with similar results for the other spin states. Here we have made use of equation (8) and the factthat

=

0 0 1 00 0 0 11 0 0 00 1 0 0

(11)z

in the usual representation. Equation (10) was derived by simply multiplying the relevantmatrices and vectors, while the same results could have been obtained more generally by usingthe properties of the Dirac matrices. Combining these results with equations (4), (7), and (9)gives

= +

++

+( )

( )kn H

q c E mc

E

p p p c

E mc22

1 (12)p

p

x y z

p

2 22 2 2 2

2 2

for the spin combination of equation (10).Inserting equation (12) into equation (6) and summing over all of the intermediate spin

states gives the correction to the photon energy as

=

+ + + + +

+

( )

( ) ( ) ( )

( )

( ) ( )p

rE

cd

E m

E mc

E

E mc E mc p c p c

E mc

1

2

1

2 2

23

(13)

( )

p G

p

p

p p

p

23

3

0

3

0 0

2 2

2

2 4 2 2 2 2 4 4

2 4

Here we have introduced the ne structure constant q c2 and the factor of ( )1 2 3 comesfrom converting the sum to an integral. The notation 0 has been used here to indicate that it isthe unperturbed photon energy 0 that appears in equation (13).

We can now use the assumption that Ep0 and m EG p to expand thedenominator in the rst term inside the integral of equation (13) in a Taylor series to rst orderin m G and to second order in 0. This gives

= + + +( )E m E

m

E E E

1

2 2

1 12

12

38

... . (14)p G p

G

p p p0

0 0

2

We have only retained terms proportional to m G in equation (14), since we are only interestedin the rst-order effects of the gravitational eld.


8

We will rst consider the effects of the last term in equation (14) and then return toconsider the remaining two terms. The contribution from the last term gives

= +

+ + + +

+

( )

( ) ( )

( )

( ) ( )E c m dp p cE

E mc

E

E mc E mc p c p c

E mc

316

23

. (15)

( )G

p

p

p

p p

p

20

0

2 2

4

2 2

2

2 4 2 2 2 2 4 4

2 4

Evaluating the integral gives

= ( )Ec

964

. (16)( ) G2 002

The velocity of light is given by = ( )c k k and the correction c to c is thus

= = ( )c k

k

E

c. (17)

( )2

0 0

Inserting the value of E ( )2 from equation (16) into equation (17) gives =cc c

964

. (18)G

02

Equation (18) is the main result of this paper. Since G is negative, this gives a small reductionin the speed of light.

Returning to the rst and second terms on the right-hand side of equation (14), it can beshown that their contributions to c are proportional to 1 02 and 1 0, respectively. These arenonphysical terms that become innite in the limit of long wavelengths, which is somewhatsimilar to the usual infrared divergences encountered in quantum electrodynamics. We canmake an intuitive argument that these terms should vanish as a result of renormalization asfollows: The loop diagram of gure 2(a) would give an innite correction to the energy of aphoton, so that the bare energy B of the photon must be innitely large as well. Identifying 0 with B would therefore cause the nonphysical terms that involve 1 B2 and 1 B to

vanish, whereas B cancels out of the nite correction of equation (17). A more rigoroustreatment of renormalization would clearly be desirable, but that may not be possible in view ofthe fact that quantum gravity appears to be nonrenormalizable.

A neutrino can also undergo a virtual process in which particles such as W bosons, Zbosons, and leptons are created, after which the virtual particles are annihilated to give back theoriginal neutrino state. The energy of the particles in the intermediate state will include theirgravitational potential energy m G, which will produce a small correction to the velocity of aneutrino that is analogous to that of a photon calculated above. But this process involves theweak interaction where the matrix elements are many orders of magnitude smaller than thosefor the electromagnetic interaction responsible for virtual electron-positron pair production. Asa result, the expected correction to the velocity of neutrinos is negligible compared to that ofphotons.


9

4. Gauge invariance and the equivalence principle

Before we consider the magnitude of this effect, it is important to note that the results of thiscalculation are not gauge invariant with respect to the gravitational eld. Conventional quantumelectrodynamics is gauge invariant only because charge is conserved via

=

jt. (19)

EE

As a result, creating an electron-positron pair in a region of uniform electrostatic potentialhas no effect on the total electrostatic energy of the system because there is no change in thetotal charge, as illustrated in gure 4(a). But the Hamiltonian of quantum electrodynamics doesnot conserve the mass of the system in a virtual state containing an electron-positron pair andthere is no equivalent of equation (19) for mass in that case, as is discussed in more detail inappendix B. Thus the creation of an electron-positron pair in a uniform gravitational potentialdoes change the energy of the system if we assume the Hamiltonian of equation (4), asillustrated in gure 4(b). This explains why the predicted change in the velocity of light in


10

Figure 4. Effects of a constant electrostatic or gravitational potential on the energy ofthe electron-positron pair produced in the Feynman diagrams of gures 2 and 3. (a) Aregion of constant electrostatic potential E is created using a uniform spherical chargedistribution with a total charge of Q. The energy of an electron-positron pair isunaffected by E because the total change in the charge is zero as required by equation(19). (b) A region of constant gravitational potential G is created using a uniformspherical mass distribution with a total mass of M . Now the energy of the electron-positron pair is changed by m2 G because the pair production process does not conservemass. There is no equivalent of equation (19) in this case.

equation (16) depends on the value of the gravitational potential and not just the gravitationaleld in violation of gauge invariance.

Based on the equivalence principle [38, 39, 56], one would expect that these effects shouldvanish in a local freely-falling reference frame. The calculations described above wereperformed using a coordinate frame that was assumed to be at rest with respect to mass M ,where it seems reasonable to suppose that the effects of gravity can be represented by theFeynman diagrams of gures 2(b) or 3. In that reference frame, the photons and neutrinoswould travel at different velocities according to equation (18). If we made a transformation to alocal freely-falling coordinate frame where the laws of physics are assumed to be the same as inthe absence of a gravitational eld, then the photons and neutrinos would be expected to travelat the same velocity. This leads to a contradiction, since there can be no disagreement as towhether or not two particles are traveling at the same velocity. Thus the Hamiltonian ofequation (4) leads to a small departure from the equivalence principle, which is closely relatedto the lack of gravitational gauge invariance noted above.

It has been predicted [57, 58] that electrons with spin up and spin down will fall atdifferent rates in a gravitational eld in apparent violation of the weak equivalence principle.(This effect is analogous to the spinorbit coupling of an electron moving in the Coulomb eldof an atom.) That may not be too surprising given that a classical object with nonzero angularmomentum will exhibit similar effects [58, 59]. But the fact that an electron is a point particlemakes this situation different from that of a gyroscope whose nite extent makes it susceptibleto tidal forces, for example. The weak equivalence principle is sometimes stated as onlyapplying to point particles with zero angular momentum, in which case it should not be appliedto electrons. This raises some questions regarding the assumptions that are inherent in thederivation of the generally covariant form of the Dirac equation. In any event, the predictedcorrection to the speed of light from equation (18) provides further motivation for experimentaltests of the equivalence principle.

5. Comparison with experimental observations

The rst neutrinos from Supernova 1987a arrived 7.7 h before the rst photons. The currently-accepted interpretation [26] of this data is that the rst burst of neutrinos must not have beenassociated with the supernova because there is no conventional explanation for how theneutrinos could have arrived at that time. If equation (18) is valid, it could explain this long-standing anomaly.

The value of cG 2 is needed in order to compare the predicted correction to the speed oflight with experimental observations such as those from Supernova 1987a. The Newtoniangravitational potential from an object with mass M at a distance R is given by

= c

GM

Rc(20)G2 2

where G is the gravitational constant. Table 1 shows the approximate value of cG 2 and thecorresponding correction to the speed of light from equation (18) for the case in which thesource of the gravitational potential is the Earth, the Sun, or the Milky Way galaxy. It can beseen that the contributions to the gravitational potential from the Earth and Sun are negligiblecompared to that of the Milky Way galaxy.


11

The gravitational potential GU from the Universe as a whole is not given by theNewtonian formula of equation (20). Instead, = 0GU for a at universe as is discussed inappendix A. Astronomical observations indicate that the Universe is at to within theexperimental uncertainty, in which case the only contribution to the gravitational potential isfrom local variations in the mass density, such as the Milky Way galaxy. Mass variations atlarger distances appear to be negligible in comparison.

Supernova 1987a was located in the Large Magellanic Cloud [60], which is a smallergalaxy that is gravitationally bound to the Milky Way galaxy. In order to predict the expecteddifference in the arrival times of photons and neutrinos at the Earth, it is necessary to integratethe effects of equation (18) over their path which is illustrated in gure 5. Longo [61] integratedthe usual relativistic factor of ( )r c2 G 02 in equation (1) over the path illustrated in gure 5using a model for the gravitational potential produced by the Milky Way galaxy. (Thecontribution of the Large Magellanic Cloud to the gravitational potential is negligible due to itssmall mass.) He obtained a total time delay of 3506 h from the usual correction to the speed oflight in equation (1). A similar calculation by Krauss and Tremaine [62] gave a total time delayof 3944 h.

The integral of equation (18) over the same path differs from these estimates by a factor of9 64 and also by a factor of 1 2, since the factor of 2 in equation (1) does not appear inequation (18). Applying this factor to the average of the results of Longo [61] and of Krauss andTremaine [62] gives a predicted delay of 1.9 h for the photons relative to the neutrinos based onequation (18). This estimate is really a lower bound on the actual delay, since refs. [61] and [62]only included the mass of the Milky Way that is within 60 kpc of the center of the galaxy. Thatrepresents roughly half of the estimated mass of the galaxy and the predicted delay could be as


12

Figure 5. Path followed by the neutrinos and photons from Supernova 1987a, whichwas located in the Large Magellanic Cloud [62]. [61] and [62] estimated the time delayexpected from equation (1) using the gravitational potential from the Milky Waygalaxy, which can then be used to calculate the contribution from equation (18).

Table 1. Gravitational potential cG 2 and fractional correction to the speed of lightc c0 from the Earth, the Sun, and the Milky Way galaxy. The value of the gravitationalpotential from the Milky Way galaxy was approximated at the location of the Earthusing equation (20).

cG 2 c c0Earth 6.4 10 10 6.6 10 13Sun 9.9 10 9 1.01 10 11Galaxy 4.2 10 6 4.3 10 9

large as 4 h if the additional mass were included. (The effects of dark matter appear to beincluded in [61] and [62] and no correction for that is required.)

The observations made during Supernova 1987a are illustrated in gure 6, which is basedon a review article by Bahcall and his colleagues [26]. A burst of neutrinos was observed by adetector underneath Mont Blanc followed 4.7 h later by a second burst of neutrinos that wasdetected in the Kamiokande II detector in Japan and the IMB detector in Ohio. The rstobservation of visible light from the supernova was then observed approximately three hoursafter the second burst of neutrinos, or 7.7 h after the rst burst of neutrinos. As mentionedearlier, the usual interpretation of this data is that the rst burst of neutrinos must not have beenassociated with the supernova for the reasons described below.

A numerical simulation of the collapse of the progenitor star gave a predicted visible lightintensity as a function of time (light curve) that is represented by the solid line in gure 6. Thereis an expected time delay of approximately three hours between the collapse of the core and theproduction of visible light at the surface of the star due to the propagation of a shock wavethrough the stellar material. (Any light produced in the interior of the star will be preventedfrom immediately reaching the surface due to diffusion). As a result, Bahcall and his colleagueshave stated that the arrival time of the rst burst of neutrinos is not consistent with the observedlight curve [26]. In addition, the fact that the rst burst of neutrinos was only detected by theMont Blanc detector and not the other two detectors, which were assumed at the time to have


13

Figure 6. Sequence of events observed during Supernova 1987a [26]. The time at whichthe rst burst of neutrinos was observed in the Mont Blanc detector is indicated by thedashed line, while the time at which the second burst of neutrinos was observed in theKamiokande II and IMB detectors is indicated by the dotted-dashed line. The datapoints show the magnitude (logarithmic intensity) of the observed visible light from thesupernova as a function of the time (in days) after the arrival of the second burst ofneutrinos. The solid line is the result of a numerical calculation based on the acceptedmodel of the supernova. The rst burst of neutrinos was considered to be inconsistentwith the accepted model and was rejected as a statistical outlier [26]. This discrepancycould be explained if the arrival of the photons was delayed as predicted by equation(18).

higher sensitivities, further suggested that the rst burst of neutrinos must have been ananomaly that was not associated with Supernova 1987a [26].

The probability that the detection of the initial burst of neutrinos in the Mont Blancdetector was a random occurrence has been estimated to be less than 10 4 [27]. As a result, thereare some experts in the eld who consider the origin of the rst burst of neutrinos to be an openquestion [27, 6365]. A more recent numerical simulation [65] showed that a progenitor starwith a sufciently high rate of angular rotation would be expected to produce an initialincomplete collapse of the core followed by a second collapse, which would produce two burstsof neutrinos instead of just one. In addition, the simulation showed that different kinds ofneutrinos with different energy ranges should have been produced during the two collapses[64, 65]. The material used in the Mont Blanc detector was different from that used in the othertwo detectors and the expected sensitivity of detection for the kind of neutrinos in the rst bursthas been estimated to be a factor of 20 higher in the Mont Blanc detector than the otherdetectors, which is consistent with the observations [64].

The possibility of a double collapse of the core suggests an alternative explanation for theobservations associated with Supernova 1987a. In this scenario, the rst burst of neutrinossignaled the initial collapse of the core with an associated production of visible light roughly 3 hlater as expected from the models. If the photons were delayed by an additional 4.7 h by thegravitational potential in equation (18), then the light would have arrived 7.7 h after the rstneutrino burst, as observed. The second collapse of the core would have produced an increase inthe intensity of the visible light approximately 4.7 h after the arrival of the rst photons. This isconsistent with the observation that the light signal increased more rapidly than would haveotherwise been expected during that time interval [26].

The photon delay of 1.9 h relative to the neutrinos as predicted by equation (18) is only40% of the 4.7 h delay assumed in the scenario described above. As mentioned earlier, thisestimate is a lower bound on the actual delay which could be as large as 4 h. Thus equation (18)is in reasonable agreement with the experimental observations and it provides a possibleexplanation for the rst burst of neutrinos that is inconsistent with the conventional model of thesupernova.

The Hamiltonian of equation (4) does not appear to be ruled out by the results of existinghigh-precision tests of quantum electrodynamics [66]. It would result in a small correction tothe anomalous magnetic moment of the electron, for example, that is much smaller than theprecision of the current experiments [28, 29]. The model would also predict [66] a correction tothe decay rate of orthopositronium that is approximately two orders of magnitude smaller thanthe current experimental precision [67]. Future experiments of that kind may eventually allowan independent test of the implications of including the gravitational potential in theHamiltonian.

6. Summary and conclusions

A simple model has been considered here in which it was assumed that, to a rst approximation,the effects of a weak gravitational eld on a quantum system can be represented by includingthe gravitational potential energy of any massive particles in the Hamiltonian. When applied tothe Hamiltonian of quantum electrodynamics, this results in virtual electrons and positronshaving a gravitational potential energy that is the same as that of a real particle. Perturbation


14

theory was then used to show that such a model predicts a small reduction in the speed of lightwhile the corresponding effects for neutrinos are negligibly small due to their weak interactions.

The predicted correction to the speed of light depends on the gravitational potential and notthe gravitational eld. An observable difference between the velocity of photons and neutrinosthat depends only on the gravitational potential is not gauge invariant. The origin of this lack ofgauge invariance can be understood from the fact that the gravitational potential energy has thesame sign for the virtual electrons and positrons created during pair production, while theirelectrostatic potential energies have the opposite sign and cancel out, as illustrated in gure 4.

The predictions of this simple model are also in disagreement with the analogouscalculations performed using the generalization of the Dirac equation to curved spacetime,which gives a much smaller effect that does depend on the gravitational eld and not thepotential itself [1719]. The lack of gauge invariance and the disagreement with the generally-covariant form of the Dirac equation both suggest that this simple model must be nonphysical.

Nevertheless, the predictions of this model are in reasonable agreement with theexperimental observations from Supernova 1987a, in which the rst neutrinos arrived 7.7 hbefore the rst photons. There is no conventional explanation for how that could have occurredand the currently-accepted interpretation is that the rst burst of neutrinos must not have beenrelated to the supernova [26], despite the fact that the probability of such an event occurring atrandom has been estimated to be less than 10 4 [27]. The correction to the speed of light fromequation (18), if correct, would explain this anomaly.

The differences between this simple model and the generally covariant form of the Diracequation are closely related to the role of the equivalence principle, since photons and neutrinosshould travel at the same velocity in a local freely-falling reference frame and thus in allreference frames. (The rest mass of a high-energy neutrino is negligible in this regard.) There isalready considerable interest in experimental tests of the equivalence principle and the results ofthe model considered here may provide further motivation for experiments of that kind.

Quantum mechanics and general relativity are two of the most fundamental laws ofphysics. Combining these two theories in a consistent way is currently one of the major goals ofphysics research. The predicted correction to the speed of light in a gravitational potential maybe of further interest if the currently-accepted principles of quantum mechanics and generalrelativity are eventually found to be incompatible in some way.

Acknowledgements

I would like to acknowledge stimulating discussions with W Cohick, A Kogut and D Shortle.

Appendix A. Gravitational potentials in general relativity

This appendix briey reviews the analogy between general relativity and electromagnetism fora weak gravitational eld, which leads to the introduction of the gravitational analogs of theelectromagnetic vector and scalar potentials. The eld equations of general relativity arenonlinear but they can be linearized if the gravitational eld is sufciently small[33, 38, 39, 4249]. In that case, we can write the metric tensor g in the form


15

= + g h (A.1)where is the diagonal metric of special relativity with elements of 1 and h is assumed tobe small. It will also be convenient to dene h and h by

h h

h h h12

. (A.2)

We can then dene [4345, 49] the gravitational vector and scalar potentials AG andG by

=

=

c h

A c h

14

14

. (A.3)

G

Gi i

200

20

We can also dene two vectors EG and BG by

=

=

E A

B Ac t

1

. (A.4)

G G G

G G

Einsteins eld equations can then be used to show that

= =

=

=

E

B

EB

BE

j

G

c t

c t

G

c

4

0

1

1 4, (A.5)

G G

G

GG

GG

G

where Gand j

Gare the mass density and current. The potentials can be also be shown to obey

the wave equations

=

=A A j

c tG

c t

G

c

14

1 4. (A.6)

GG

G

GG

G

22

2

2

22

2

2

Equations (A.4) through (A.6) are the same as those of classical electromagnetism exceptfor the factor of G and the sign of the source terms.

The geodesic equation can also be used to show that the trajectory of a particle of mass mis given [4345, 49] by

= = + r f E r Bmt

m mc t

d

d4

1 dd

(A.7)G G G

2

2

in the limit of low velocities. Here fGis the gravitational force and equation (A.7) is the same as

the Lorentz force in electromagnetism except for the factor of 4. Some authors redene a newvector potential =A A 4G G and a new gravitomagnetic eld =B B 4G G in order to put


16

equation (A.7) into the same form as in electromagnetism. In that case the wave equation of(A.6) no longer holds in its present form. This factor of 4 is due to the fact that the gravitationaleld is a tensor and not a vector, and its appearance somewhere in the equations is unavoidable.

One of the most interesting and fundamental features of the quantum theory is the fact thatit is the electromagnetic potentials AE and E, not the electromagnetic elds EE and BE, thatappear in the Hamiltonian for a charged particle [53]. This gives rise to the AharonovBohmeffect [50] in which there are observable phenomena that occur in regions of space where

=E 0E and =B 0E , for example. This and the Lorentz force equation (A.7) both suggest thatthe Hamiltonian for a nonrelativistic particle in a weak gravitational eld should be taken to be

=

+AHm i c

m m12

4(A.8)G G

2

in analogy with electromagnetism. Equation (3) in the text results from replacing the vectorpotential with =A A 4G G . Equation (A.8) has previously been used in connection with thegravitational analog of the AharonovBohm effect [33, 47, 5052], for example.

It can be seen from equation (A.3) that the gravitational potentialGU from the Universe asa whole is zero for a at universe where =h 0 aside from the effects of local mass densityvariations. This justies the assumption in the main text that the total mass of the Universe doesnot contribute to the correction to the speed of light as might be expected from the Newtonianexpression of equation (20).

Appendix B. Mass density operator and the nonrelativistic limit

It was assumed in the text that the gravitational potential energy of a virtual electron-positronpair is m2 G while the electrostatic potential energy cancels to zero. This difference between ( )rG and ( )rE is responsible for the predicted correction to the speed of light as well as thelack of gauge invariance in gure 4. The purpose of this appendix is to dene a mass densityoperator ( )rG with these properties. The nonrelativistic limit of the interaction Hamiltonian ofequation (4) is then shown to reduce to the Pauli equation with the correct sign of thegravitational potential energy for both electrons and positrons.

The electric charge and current densities can be written in covariant form as a four-vector

= ( ) ( ) ( )r r rj qcE , whose fourth component is c times the charge density = ( ) ( ) ( )r r rqE . Here the

are the usual Dirac matrices and ( ) ( )r r is theadjoint eld operator, where is given as usual by

=

1 0 0 00 1 0 00 0 1 00 0 0 1

. (B.1)

Roughly speaking, the minus signs in ensure that a positron will have the opposite chargefrom an electron. Since we want the gravitational potential to have the same sign for bothelectrons and positrons, this suggests that the simplest choice for ( )rG may be


17

= ( ) ( ) ( )r r rm . (B.2)GThis differs from ( )rE by the absence of which would be expected to give a positivegravitational potential for both electrons and positrons.

We will rst investigate the nonrelativistic limit of the interaction Hamiltonian in

equation (4). Combining H with the remaining noninteracting terms in the Hamiltonian andinserting ( )rG from equation (B.2) gives the total Hamiltonian H :

= + + +

+ +

( )

( )

AH d r ci

q

cmc m q

a a 1 2 . (B.3)k

k k k

G E3 2

,, ,

This is the standard Hamiltonian of quantum electrodynamics [68] aside from the m G term.The time dependence of ( )r t, in the Heisenberg picture can be calculated using the anti-commutation property = { }( ) ( ) ( )r r r rt t, , , 3 , which gives [68]

=

= + +

+

( )Ait i

H ci

q

cmc m

q

1,

. (B.4)

G

E

2

This is the usual Dirac equation for the second-quantized eld operator with the addition of thegravitational potential term.

The nonrelativistic limit of equation (B.4) can now be calculated using the approachdescribed in ref. [53], for example. We rst rewrite the four-component eld operator in theform

=( )( )( )

rr

rt

t

t,

,

,(B.5)

where ( )r t, and ( )r t, are two-component spinors. The Dirac equation (B.4) is thenequivalent to

= + + +

= +

( )

( )

A

A

it

ci

q

cq mc m

it

ci

q

cq mc m , (B.6)

E G

E G

2

2

where denotes the Pauli spin matrices. If we consider a positive-energy eigenstatecorresponding to an electron with a velocity v c, then when acting on that state andthe second line of equation (B.5) gives to lowest order

=

Amc i

q

c

12

. (B.7)

Here we have assumed that the potential energies are small compared to the rest mass and thatthe time rate of change of the state is imc2 to lowest order [53]. Inserting equation (B.7) intothe rst line of equation (B.6) and using the identity


18

= + ( ) ( ) ( ) ( )a b a b a bi (B.8)gives

= + + + ( )A Bi

t m i

q

c

q

mcq mc m

12 2

. (B.9)E G

22

This is the usual Pauli equation written in terms of (nonrelativistic) second-quantized eldopertors [53] with the addition of the gravitational potential term.

If we consider an eigenstate corresponding to a positron instead, then when actingon that state and the rst line of equation (B.6) gives to lowest order

=

Amc i

q

c

12

. (B.10)

Inserting this into the second line of equation (B.6) gives

= +

+ ( )

A Bit m i

q

c

q

mc

q mc m

12 2

. (B.11)E G

2

2

This can be rewritten by dening a new operator = (charge conjugation) and taking theadjoint of equation (B.11), which gives

= + + + + ( )A Bit m i

q

c

q

mcq mc m

12 2

. (B.12)E G

22

Equation (B.12) corresponds to the Pauli equation for a particle (a positron) whose charge andspin are opposite that of an electron but whose mass and gravitational potential energy are thesame as that of an electron. Taking = =A 0E gives the nonrelativistic Schrodinger equationof equation (2) as desired. Equations (B.9) and (B.12) show that this choice of ( )rG gives thecorrect sign of the gravitational potential energy for both electrons and positrons at least in thenonrelativistic limit.

We now generalize this to the case in which the virtual electron-positron pair may haverelativistic velocities. Here we make use of the fact that, in the text, the gravitational potential

was included in the unperturbed Hamiltonian H0, where = + H H H0 . The perturbationcalculations were then based on the eigenstates of H0 and their corresponding energyeigenvalues E0, which were assumed to include a gravitational potential energy of m G for eachparticle. First consider the value of the eigenvalue E0 for a positron state = c 0ks withmomentum k and spin s, where 0 is the vacuum. For a weak eld with c 1G 2 , thegravitational contribution EG to the eigenvalue E0 is given [53] to lowest order in perturbationtheory in m G by

= = ( ) ( )E H c d rm r r c0 0 . (B.13)k kG s G s3

Here we have used the gravitational potential term in the Hamiltonian of equation (4) andinserted the denition of ( )rG from equation (B.2). Using the form of ( )r from equation (7)gives the relevant terms in ( )rG as


19

=

=

+

( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

p p p p

p p

p p

mmc

Ec s v s e

mc

Ec s v s e

mmc

E

mc

Ev s v s e e

c s c s

, , , ,

, ,

, , . (B.14)

p p

p r

p p

p r

p p p p

p r p r

pp

Gs

i

s

i

s s

i i

ss

,

2

,

2

, ,

2 2

,

Here the order of the operators ( )pc s, and ( )pc s, were interchanged using their anti-commutator. The term involving the Kronecker delta function is a constant (nonoperator) that isindependent of the state of the system and can be ignored [68]. (This procedure is alsonecessary in order to obtain the correct charge of a positron.) Inserting equation (B.14) intoequation (B.13) gives

= =

=

( ) ( )( ) ( ) ( ) ( )k k k kE mcE

v s v s mmc

Ev s v s m

mc

Em

, , , ,

. (B.15)

k k

k

G G G

G

2 2

2

The right-hand side of equation (B.15) follows from the fact that = ( ) ( )k kv s v s, , 1[12, 14, 15].

It can be seen from equation (B.15) that a positron will have a gravitational potentialenergy with the correct sign but multiplied by a relativistic factor that depends on the value of k.This can be avoided if we dene a new set of spinors ( )ku s, and ( )kv s, that are dened by

=

=( ) ( )( ) ( )

k k

k k

u s E mc u s

v s E mc v s

, ,

, , . (B.16)

k

k

2

2

We also dene the operator ( )r by

= + ( ) ( ) ( ) ( )r p p p pmcE b s u s e c s v s e( ) , , , , (B.17)p pp r p r

s

i i

,

2

where the spinors of equation (B.16) have been inserted into equation (7). The denition of ( )rG in equation (B.2) is now replaced by

= = ( ) ( ) ( ) ( ) ( )r r r r rm m , (B.18)Gwhile the usual eld operator ( )r is used in ( )rE and ( )j rE .

With this choice of ( )rG , equation (B.15) becomes =E m (B.19)G G

and the energy of a positron in a gravitational potential is +E mk G as desired. The same resultcan also obtained for the energy of an electron. This justies the assumption in the text that avirtual electron-positron pair has a gravitational potential energy of m2 G.


20

The nonrelativistic limit is not affected by this new denition of ( )rG because ( ) ( )r r in that limit and equations (B.9) and (B.12) remain valid. It can also be shownthat the expectation value in the state of the integral of ( )rG over all space is equal to themass m as would be expected.

It should be emphasized that the model described in this paper is only intended to providean alternative and approximate description of the propagation of photons in a gravitationalpotential; it is not intended to represent a complete or consistent theory. For example, ( )rGcannot obey a conservation law as a result of pair production and it is not part of a covariantfour-vector. A more rigorous discussion of related issues in the currently-accepted formulationof the Dirac equation in curved spacetime will be submitted for publication elsewhere.

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1. Introduction2. Gravitational potentials in the quantum theory3. Calculated correction to the speed of light4. Gauge invariance and the equivalence principle5. Comparison with experimental observations6. Summary and conclusionsAcknowledgementsAppendix A.Appendix B.References

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Apparent correction to the speed of light in a gravitational potential.pdf