aa r X i v : . [ phy s i c s . g e n - ph ] J a n Electrostatic time dilation andredshift
Murat ¨Ozer ∗ Yıldız Teknik ¨Universitesi, Fizik B¨ol¨um¨u, 34220 Esenler, ˙Istanbul, Turkey&TUSAS¸, YT ¨U Yıldız Teknopark, 34220 Esenler, ˙Istanbul, Turkey
Abstract
We first present the salient features of the gravitational time dilationand redshift effects in two ways; by considering the oscillation frequen-cies/rates of clocks at different heights/potentials and by consideringthe photons emitted by these clocks such as atoms/nuclei. We thenpoint out to the extension of these gravitational effects to static elec-tricity along with two experiments performed in the ’30s with nullresults of the electrostatic redshift. We show that the absence of thisredshift is a consequence of the conservation of electric charge. Wediscuss the electrical time dilation and redshift effects in detail andargue that the electrostatic time dilation in an electric field must bea fact of Nature. We then present a general relativistic scheme thatexplains this effect. We also introduce an electrical equivalence princi-ple analogous to the gravitational one and demonstrate how to obtainthe electrostatic time dilation by this principle. We emphasize theimportance of ionic atomic clocks to measure this effect whose confir-mation would support the general relativistic scheme presented. Wefinally go over an attempt in the literature to explain the impossibilityof the experimental observation of the electrostatic redshift due to itssmallness by employing the Reissner - Nordstr¨om metric in generalrelativity. We argue that the Q - term in this metric is due to theminuscule contribution of the energy of the electric field of the centralbody to its gravitational field. Thus being gravitational, this metriccannot be used to calculate the amount of the alleged electrostatic ∗ [email protected] edshift. Keywords:
Gravitational time dilation and redshift, Pound - Rebka - SniderExperiment, Electrical time dilation and refshift, Ionic atomic clocks, Reiss-ner - Nordstr¨om metric
That time passes differently at different heights or potentials in a gravita-tional field is called the gravitational time dilation. The time intervals be-tween two events measured by observers located at different altitudes froma gravitational source, a large mass, happen to be different. Time passesfaster, in other words, the rate of a clock, namely its oscillation frequency,increases as it gets located farther from a gravitating source. The effect wasfirst predicted by Einstein [1, 2] and was experimentally verified indirectlyby means of the M¨osbauer effect in [3, 4] and directly in [5, 6] by readings ofthe airborne and earthbound atomic clocks, and in [7, 8] by comparing thefrequencies of microwave signals from hydrogen maser clocks in a rocket ata high altitude and at an Earth station. The precision of these experimentswere improved recently by measuring the frequencies of the onboard hydro-gen maser clocks in Galileo Satellites of the European Space Agency [9, 10].It was reported in [11] that time dilation due to a change in height less thana meter could be detected by comparing two optical clocks based on Al + ions.A directly related concept is the redshift of light in a gravitational field[29]. Usually, it is defined as the lengthening (shortening) of the wavelength(frequency) of light as it moves away from a massive body such as the Earth.Light is thought to loose energy as it reaches higher altitudes in a gravita-tional field due to its interaction with the field [12, 13, 14, 15]. However, asit has been pointed out in [16] and emphasized in [17], light does not actu-ally interact with the gravitational field and lose energy as it moves up tohigher potentials. This is because light, strictly speaking, does not have agravitational mass and thereby cannot be assigned a potential energy. Whathappens in reality is that the frequency of light emitted by an atom (or anucleus) at a lower gravitational potential is smaller than the frequency oflight emitted by an identical atom/nucleus at a higher potential [17].Certain similarities between classical gravitational and electric fields leadone naturally to question if a similar effect might take place in a static electric2eld. Long before the experimental confirmation of the gravitational effect,this possibility was exercised and found that light does not undergo an actualfrequency change as it moves in a static electric field from a low potentialto a higher one or vice versa [18]. An interferometer was used to comparethe frequencies of light before and after traveling in a static electric field. Asecond experiment ended up with the same null conclusion [19]. There was noattempt in these works to define and discuss the electrostatic redshift of light .An endeavor to explain the null results of these experiments theoreticallywas presented in [20] by employing the Reissner-Nordstr¨om metric in generalrelativity.The purpose of the present work is to discuss, to our knowledge for thefirst time in the literature, the effects that occur in a static electric fieldsimilar to those of the gravitational ones. To this end, we first expound thegravitational effects and then extend them to their electrical analogues. Wepresent a general relativistic scheme that predicts the electrical effects too.We demonstrate that the electrostatic effects can also be obtained from an electrical equivalence principle . We propose an experiment with two Al + ions at different potentials in an electric field to measure the electrostatic timedilation effect. Finally, we argue that the Reissner - Nordstr¨om - treatmentof the electrostatic redshift in [20] to explain the null results of the electricalredshift experiments reported in [18, 19] involves a conceptual error and isirrelevant to electrical redshift. Let us consider an atom of mass m at a height H in a static gravitationalfield like that of the Earth. Taking this height as the reference level for thegravitational potential, the energy of the atom is E (0) = m i c , (1)in its ground state, and E ∗ (0) = m i c + E = m ∗ i c , (2)in an excited state, denoted by a superscript ∗ , whose energy exceeds that ofthe ground state by E . Raising the atom by a height H changes the energylevels to E ( H ) = m i c + m p gH, (3)in the ground state, and to E ∗ ( H ) = m ∗ i c + m ∗ p gH, (4)3n the excited state, where g ≈ . m/s is the local value of the gravitationalacceleration on the surface of the Earth, m i and m p are the inertial andpassive gravitational masses. Here m ∗ i and m ∗ p are the excited masses of theatom and are given by m ∗ i = m i + E c .m ∗ p = m p + E c . (5)As a result of this raising, the oscillation frequency f = E/h of the atomchanges from f (0) to f ( H ) in the ground state, and from f ∗ (0) to f ∗ ( H )in the excited state, where h is the Planck constant. These frequencies atdifferent altitudes are related to each other in the ground and excited states,respectively as f ( H ) = f (0) (cid:18) m p m i gHc (cid:19) = f (0) (cid:18) gHc (cid:19) ,f ∗ ( H ) = f ∗ (0) (cid:18) m ∗ p m ∗ i gHc (cid:19) = f ∗ (0) (cid:18) gHc (cid:19) , (6)where we have used m p /m i = m ∗ p /m ∗ i = 1. Therefore, the fractional changesin the oscillation frequencies due to this raising are given by δff (0) = f ( H ) − f (0) f (0) = gHc ,δf ∗ f ∗ (0) = f ∗ ( H ) − f ∗ (0) f ∗ (0) = gHc , (7)which are the same both in the ground and excited states. Notice that gH inthese equations and what follows is equal to the change ∆ φ g = φ g ( H ) − φ g (0),where φ g is the gravitational potential. We note that these fractional changesare equal to the ratio of the work done against gravity by the external agentin raising the masses m and m ∗ by H , to their rest energies. Taken as afrequency reference, such an atom can be considered as a clock whose rate,its oscillation frequency, is faster the higher the height H is. Thus, thetime intervals measured by such a clock are proportional to its oscillationfrequencies and are given by [30]∆ T ( H ) = ∆ T (0) (cid:18) m p m i gHc (cid:19) = ∆ T (0) (cid:18) gHc (cid:19) , ∆ T ∗ ( H ) = ∆ T ∗ (0) (cid:18) m ∗ p m ∗ i gHc (cid:19) = ∆ T ∗ (0) (cid:18) gHc (cid:19) , (8)4hich indicate how such atoms/clocks age. Thus we find out that atoms athigher altitudes, or higher potentials, age faster than those at lower altitudesfrom the gravitating source by a fractional change δ ∆ T ∆ T (0) = δ ∆ T ∗ ∆ T ∗ (0) = gHc = ∆ φ g c . (9)As we will now show, the same results can be obtained by considering thephoton emissions in the excited states. An excited atom at height 0 whoseenergy is as in Eq.(2) may make a transition to the ground state by emittinga photon of frequency ν (0) = E ∗ (0) − E (0) h = E h . (10)When this photon is directed up, it cannot be absorbed by an atom at height H in its ground state whose energy is as in Eq.(3) due to the energy deficiency.An energy of E + E gH/c is required of the photon to excite the atom so thatits energy has the value in Eq.(4). As has been emphasized in [17], the photonwith energy E does not loose energy as it moves up in the gravitational field.In the experiment [3, 4], the extra energy E gH/c the photon needs to haveso as to be absorbed is supplied to it through the Doppler effect. The source,the iron - 57 nucleus at height 0, is moved up toward the absorber to increasethe energy of the photon to E γ = E (cid:18) vc − vc (cid:19) / ≈ E + E vc . (11)This is achieved by adjusting the upward speed of the source, as in [3, 4], to v = gHc . (12)As is pointed out in [17], had the photon lost energy as it moved up, as isassumed in the wrong interpretation of the gravitational redshift, the requiredDoppler speed would have been twice as that in Eq.(12) [31]. The excitednucleus at H with enegy E ∗ ( H ) may then make a transition to its groundstate. The frequency of the photon emitted in this transition is ν ( H ) = E ∗ ( H ) − E ( H ) h = E + E gH/c h = ν (0) (cid:18) gHc (cid:19) , (13)5hich is naturally equal to the frequency of the photon absorbed. The frac-tional change δν/ν (0) in the frequencies of the photons emitted and thepredicted aging of the nuclei at different altitudes are, respectively, the sameas those in Eq.(7) and Eq.(8).One may wonder why the photon is still said to undergo a redshift eventhough its energy/frequency does not change as it moves up or down in agravitational field. As is emphasized in [17], this is because the energy levelsof atoms/nuclei at higher altitudes undergo a blueshift, namely their energiesincrease, relative to atoms/nuclei at lower altitudes. The photon emitted by anucleus/atom at a lower altitude and absorbed by an identical atom/nucleusat a higher altitude is seen by this atom/nucleus to be redshifted. This canbe understood as follows: The higher-up-atom/nucleus, the clock, measuresthe frequency ν (0) of the photon when it reaches it as ¯ ν ( H ), given by¯ ν ( H ) = ¯ ν (0) (cid:18) ν (0) ν ( H ) (cid:19) = ν (0) (cid:18) − gHc (cid:19) , (14)with a fractional change of − gH/c , and ¯ ν (0) = ν (0).To sum up what we have reviewed so far, our analysis shows clearly that(i) the gravitational time dilation effect is due to the difference in the totalenergies of objects when they are at different altitudes in a gravitational field,(ii) the frequency of photons do not change as they move in a gravitationalfield, and (iii) atoms/nuclei higher up in a gravitational field whose energiesare blushifted see the photons emitted by atoms/nuclei lower down in thefield as redshifted. As is well known, the gravitational time dilation effect can be obtained fromthe Schwarzschild line element ds = − (cid:18) − GMc r (cid:19) c dt + (cid:18) − GMc r (cid:19) − dr + r dθ + r sin θdφ , (15)where G is the gravitational constant, M the mass of a spherical body, and c the speed of light. The proper time interval dτ for a test particle located atthe coordinate r is related to its coordinate time interval dt by the relation dτ = (cid:18) − GMc r (cid:19) / dt. (16)6he ratio of the proper time intervals for two clocks (test particles) at r = R + H and r = R is thus dτ ( R + H ) dτ ( R ) = ∆ T ( H )∆ T (0) = (cid:18) − GMc ( R + H ) (cid:19) / . (cid:18) − GMc R (cid:19) / , (17)because the two coordinate time intervals are equal. Since the height H ofthe top clock is much smaller than the radius R of the Earth and the termswith c − are much smaller than 1 the above ratio is approximated by∆ T ( H )∆ T (0) ≈ (cid:18) gHc (cid:19) , (18)where g = GM/R . This is the same expression as that in Eq.(8). Thus thegravitational time dilation is indeed a prediction of the Schwarzschild lineelement. Next, we undertake the electrical analogues, if they exist, of the gravitationaleffects. To this end, let us consider an electric field E in a region of the xy plane directed from right to left in the − x axis. Assume there are four identi-cal positively charged atoms (cations) in their ground states at a level on the x axis where the electrical potential will be taken to be zero. The atoms areprevented from interacting with each other through some mechanism. Theenergies of the atoms at this zero level of potential are given by Eq.(1), asin the gravitational case. Then we move one of the atoms to the right by adistance d on the x axis. The energy of this atom is [32] E ( d ) = m i c + q | E | d, (19)where q is the positive charge of the atom. Then, we excite two of the atomsby letting them absorb a photon of energy E increasing their energy to E ∗ (0) = m i c + E . (20)Afterwards, we move one of these excited atoms to the right by the samedistance d as before. The energy of this atom will be E ∗ ( d ) = m i c + E + q | E | d. (21)The fractional changes in the energies and vibrational frequencies are δEE (0) = δff (0) = qm i | E | dc = qm i ∆ φ e c , (22)7n the ground state, and δE ∗ E ∗ (0) = δf ∗ f ∗ (0) = qm i | E | dc (cid:16) E m i c (cid:17) ≈ qm i | E | dc = qm i ∆ φ e c , (23)in the excited state where E /mc ≪ φ e = | E | d > T ( d ) = ∆ T (0) (cid:18) qm i | E | dc (cid:19) , ∆ T ∗ ( d ) = ∆ T ∗ (0) (cid:18) qm i | E | dc (cid:19) , (24)which are similar to the gravitational ones in Eq.(8). It should be noted thatthese electrical time dilation effects become contraction effects for anions,atoms with a total negative electric charge, for ∆ φ e >
0. This is an effectwith no counterpart in the gravitational case due to the absence of negativemass. The excited atoms at x = 0 and at x = d can make a transition totheir ground states by emitting a photon of frequency ν (0) = E ∗ (0) − E (0) h = E h ,ν ( d ) = E ∗ ( d ) − E ( d ) h = E h . (25)It is no surprise that these frequencies are equal. This is a consequence of theconservation of electric charge. Moving an electric charge from one point toanother in an electric field by a distance d does not change the amount of theelectric charge whereas moving it vertically by a height H in a gravitationalfield would change the gravitational mass by gH/c .Let us contemplate a Pound - Rebka - Snider experiment [3, 4] performedin an electric field. The photon of energy E emitted by the source/emitter at x = 0, whose energy is as given in Eq.(20), will reach the absorber/receiveratom, which is in its ground state at x = d whose energy is as given in Eq.(19).The photon can be absorbed by this atom without the need to increase itsenergy by a Doppler shift because the mass of the atom plays no role inthe energy shift in an electric field in contrast with the gravitational case.Having absorbed the photon, this excited atom can in turn emit the photonwith the same energy E by making a transition to its ground state. Theoverall result is null. There is no change in the frequencies of the absorbed8nd emitted photons. Thus no electrical redshift! This is also predicted bythe electrical analogue of Eq.(14), which would yield ¯ ν ( d ) = ν (0). We cannow understand the null results of the electrical redshift experiments. In[18], light of a certain frequency from an electrodeless discharge in mercuryvapor was let to travel through a potential difference of 50 kV above or belowground. A quartz interferometer kept at zero potential was used to measurethe frequency of the received light. No change in the frequency of the lightwas detected. This experiment was repeated in [19] by employing a potentialdifference of ± kV and the findings in [18] were improved by a factor often. Theoretically, these results are expected because the energy/frequencyof light does not undergo a shift as it travels between different electrostaticpotentials in an electric field, just as what happens in a gravitational field aswe have discussed. Comparing equations (8) and (24) reveals that the latter can be obtainedfrom the former on replacing m p with q , g with | E | and H with d . This couldbe interpreted as the possibility that there might exist a general relativistictheory of electromagnetism which is very similar to that of gravitation. Thistheory might be a unified theory of gravitation and electromagnetism. Itmight predict [21] the spacetime metric outside a spherical body of mass M and electric charge Q as ds = − (cid:18) − m p m i GMc r + qm i k e Qc r (cid:19) c dt + (cid:18) − m p m i GMc r + qm i k e Qc r (cid:19) − dr + r dθ + r sin θdφ , (26)where k e is the Coulomb constant and m p /m i = 1. It just follows that theproper and coordinate time intervals of a test particle located at r are relatedas dτ = (cid:18) − GMc r + qm i k e Qc r (cid:19) / dt. (27)Let us contemplate an experiment with a charged metallic sphere with aradius R of few decimeters. Assume that the sphere is negatively charged ( Q = −| Q | ) so that its electric field is directed from the right to the left alongthe horizontal towards the equator of the sphere. The ratio of the propertime intervals for two clocks (test particles) located on the equatorial plane9t r + d and r for r > R is thus dτ ( r + d ) dτ ( r ) = ∆ T ( d )∆ T (0) = (cid:18) − qm i k e | Q | c ( r + d ) (cid:19) / . (cid:18) − qm i k e | Q | c r (cid:19) / , = (cid:18) qm i c φ e ( r + d ) (cid:19) / . (cid:18) qm i c φ e ( r ) (cid:19) / (28)because the two coordinate time intervals are equal and φ e ( r ) = − k e | Q | /r isthe electric potential. There is no gravitational contribution due to the Earthbecause the clocks are at the same height. The gravitational contribution ofthe sphere due to its mass is negligibly smaller than its electrical contribution.Since the rest energy m i c of the test particle is much larger than its potentialenergy qφ e the above ratio is approximated by∆ T ( d )∆ T (0) ≈ (cid:18) qm i c ∆ φ e (cid:19) = (cid:18) qm i | E | dc (cid:19) (29)where ∆ φ e = φ e ( r + d ) − φ e ( r ) > d in the electric field E . This is the sameexpression as the first one in Eq.(24). It should be noted that even thoughthe electric field and the potential difference in Eq.(29) are due to a chargedmetal sphere, the result is general as the considerations in section 4 imply.The potential difference may be that in an electric field created by othermeans, as in section 7. Thus the metric in Eq.(26) provides a completeexplanation both for the gravitational and electrostatic time dilation effects.The exact one - to - one similarity of the electrostatic time dilation to itsgravitational counterpart both in the classical energy considerations and thegeneral relativistic treatments and the fact that the gravitational one is afact of Nature suffice it to conclude that the electrostatic effect, too, mustbe a fact of Nature. The answer to this question is affirmative. Electrostatic experiments per-formed in an accelerated frame would produce identical results as electro-static experiments, with a single charged particle or particles of the sametype, performed in a uniform electric field. To this end, let us consider acabin in a rocket in deep space where there are no fields of any kind. Let10here be two charged particles (observers) of the same type at the bottomand the top of the cabin of height d . Let us establish a Cartesian coordi-nate frame and choose the direction of motion of the rocket as the upwardz-axis. Let the bottom of the cabin and the origin of the coordinate frameoverlap at time t = 0. Let the rocket, hence the cabin, accelerate upwardat a cabin = − qm i E , where q is the electric charge of some known particle, m i is its inertial mass, and E is some known electric field directed in thenegative z direction as the gravitational field for convenience. The charge q will be taken positive in the following discussion for simplicity. Accordingly,adapting the gravitational equations in [23] to electricity (and reversing thedirection of the light pulses emitted) the positions of the bottom and topcharges (observers) will be given by z B ( t ) = 12 qm i | E | t ,z T ( t ) = d + 12 qm i | E | t . (30)The charge (observer) at the bottom emits two light pulses directed upwardat t = t and at t = t = t + ∆ τ B seperated by a time interval of ∆ τ B .Let these pulses be received by the charge at the top at times t = t ‘1 and t = t ‘2 = t ‘ + ∆ τ T separated by a time interval of ∆ τ T .The distance traveledby the first pulse is z T ( t ‘1 ) − z B ( t ) = c ( t ‘ − t ) , (31)which gives d + 12 qm i | E | t ‘12 − qm i | E | t = c ( t ‘1 − t ) . (32)Similarly, the distance traveled by the second pulse is z T ( t ‘1 + ∆ τ T ) − z B ( t + ∆ τ B ) = c ( t ‘ + ∆ τ T − t − ∆ τ B ) , (33)which gives d + 12 qm i | E | ( t ‘1 + ∆ τ T ) − qm i | E | ( t + ∆ τ B ) = c ( t ‘+∆ τ T − t − ∆ τ B ) , (34)Using Eq.(32) in Eq.(34) and setting t ‘1 = t + d/c and neglecting all termssecond order in time, we get∆ τ T ( c − qm i | E | dc ) ≈ c ∆ τ B (35)or, ∆ τ T ≈ ∆ τ B (1 + qm i | E | dc ) . (36)11his is the same as Eq.(24) above [33]. Now, the electrical equivalence prin-ciple tells us that this accelerated frame in deep space where there exists nofields of any kind is equivalent to a stationary frame (a lab) on Earth wherethere exists a uniform downward electric field and the elapsed times betweentwo emissions of photons by particles (atoms, etc.) of charge q seperated by adistance d is given by Eq.(36) above. It should be noted that the equivalenceof the two frames is realized not for a unique value, but for all values of q/m i of the particles of the same type with which the experiments are done. Thispoint is of utmost importance that it deserves to be elucidated further.To this end, consider a collection of charged particles with different q/m i ratios moving in an external electric field E . Adapting the gravitationalequations in [24] to electricity, the equation of motion of the nth particle willbe m ( n ) i d r ( n ) dt = q ( n ) E + X M F ( r ( n ) − r ( M ) ) , n = 1 , , ...., N, (37)where F denotes the interparticle interactions. The spacetime transforma-tions t ′ = t, r ′ ( n ) = r ( n ) − (cid:16) q ( n ) /m ( n ) i (cid:17) E t ′ , n = 1 , , ...., N, (38)cast Eq. (37) to m ( n ) i d r ′ ( n ) dt ′ + m ( n ) i (cid:16) q ( n ) /m ( n ) i (cid:17) E = q ( n ) E + X M F ( r ′ ( n ) − r ′ ( M ) ) m ( n ) i d r ′ ( n ) dt ′ = X M F ( r ′ ( n ) − r ′ ( M ) ) . (39)The electric force on the nth particle F ( n ) E = q ( n ) E (40)has been cancelled by the following fictitious force F ( n ) fict = − m ( n ) i (cid:16) q ( n ) /m ( n ) i (cid:17) E = − q ( n ) E . (41)In other words, F ( n ) E + F ( n ) fict = 0 . (42)As is seen clearly from Eq.(41) that there is no restriction on the ratio( q ( n ) /m ( n ) i ) for the cancelation of the external electric force on the parti-cle locally. Especially, it is not required for this cancellation that this ratio12e 1. It can be equal to any value observed in Nature. Furthermore, thecancelation of the electric field in the neighbohood of the nth particle occursindependently of the cancelation of the electric field for the other particles.Thus the electrical equivalence principle can be stated, adapting the grav-itational statement in [25] to electricity, as ”It is always possible at anyspace-time point of interest to transform to coordinates such that the effectsof electricity will disappear over a differential region in the neighborhood ofthat point, which is taken small enough so that the spatial and temporalvariation of electricity within the region may be neglected.” It can also bestated in the form ”It is impossible to distinguish the fictitious inertial forcefrom the real electrical force in a local region containing a single particle”.We reiterate, just as the equivalence of an accelerating frame in deep spaceand a frame at rest in a local uniform gravitational field, a similar equivalenceexists for electricity too. A frame containing a charged particle of mass m i and charge q and accelerating in deep space at an acceleration a = − ( q/m i ) E is equivalent to a laboratory frame at rest where there is an identical particlein it and a uniform electric field E . As has been demonstrated above, theelectricity experiments involving the charged particle in these two frames giveidentical results. As in gravity, there exists a second pair of equivalent framesthat result in by the addition of an acceleration a = ( q/m i ) E to the deepspace - frame and the Earth - frame. Thus the fame accelerating upward isreplaced by a frame at rest in deep space, irrespective of whether it is chargedor not, and the frame at rest in an electric field on Earth is replaced by acharged frame with Q/M = q/m i falling with acceleration a = ( q/m i ) E in auniform electric field. These two frames are completely equivalent as far asthe results of the experiments performed with the charged particle at hand.The effect of applying the spacetime transformation in Eq.(38) on eachparticle is equivalent to putting each particle in a small enough cabin whose Q/M i ratio is the same as the q/m i ratio of the particle in it. Each such cabinwill be a local reference frame with Cartesian coordinates. The nth cabin forexample, will be falling freely at an acceleration a ( n ) = Q ( n ) /M ( n ) i E . If theinterparticle interactions are neglected, each cabin would be a local inertialframe. The acceleration of the n th particle relative to its own hypotheticalcabin as it falls will be [21, 22] a ( n ) rel = q ( n ) m ( n ) i − Q ( n ) M ( n ) i ! E = 0 . (43)Each such cabin with a charged particle in it is equivalent to a similar cabinfloating in deep space. Though there is no need, we can group the particleswith the same q/m i ratio and put them in a larger cabin whose charge-13o-mass ratio Q/M is the same and falling freely in the electric field E .Obviously, it is not legitimate to put various charged particles with different q/m i ratios in a falling frame and require that they all float in this frame.This is because this would compel the q/m i ratios of the particles to be thesame and equal to the Q/M ratio of the falling cabin. Because any spacetimepoint contains a single particle only, the equivalence principle stated abovemay be called the single - particle equivalence principle [21, 22].
Before we pass to the next item, we point out to the experimental possi-bility of measuring the electrostatic time dilation effect by high precisionclocks similar to the atomic ones. What is required for such a measurementis ionic clocks [26, 27, 28] whose oscillators are positively (or, in principle,negatively) charged ionic atoms whose energy levels will split in an electricfield depending on their positions in the field [32]. For example, the frac-tional change in the oscillation frequencies of two Al + optical ion clocksin a static electric field for a potential difference of ∆ φ e ( inV ) between themwould be δff (0) = qm ∆ φ e c = 3 . × − ∆ φ e /V, (44)where q = e = 1 . × − C is the charge of Al + and m = 4 . × − kg its mass. This would be much larger than the special relativistic andgravitational time dilation effects reported in [11] depending on the value of∆ φ e . The experimental confirmation of the electrostatic time dilation effectwould be an unequivocal indication for the correctness of the unificationmetric in Eq.(26). Finally, we comment on the work in [20], where an attempt to explain thenull results of the electrostatic redshift experiments reported in [18, 19] wasmade, by reproducing the derivation of the general relativistic prediction ofthe electrical redshift in [20]. They start off by giving the fractional changein the frequency of light in general relativity, which is∆ ν/ν = 1 − ( g /g ′ ) / , (45)14here ∆ ν = ν observed − ν with ν being the frequency of the light emitted bythe source. g is the coefficient of the timelike coordinate in the square ofthe differential line element ds = g µν dx ν dx ν (46)at the position of the absorver/receiver, and g ′ being the one at the positionof the source/emitter. The absorber is assumed to be located on the dome ofan electrostatic accelerator (like a van de Graaf generator) whose electrostaticpotential is given by (in SI units) ϕ = k e Q/R, (47)where Q is the charge of the dome, R the radius of the dome. The g on thedome is assumed to be that of the Reissner - Nordstr¨om line element givenby g = 1 − GM/Rc + k e GQ /R c , (48)where M is the mass of the dome. The authors say they are only interestedin the electrostatic effect and thereby drop the second term in the equationabove as a result of which Eq.(45) becomes∆ ν/ν = 1 − (cid:0) Gϕ /k e c (cid:1) / , (49)where g ′ = 1 because the source is assumed to be in a region of zero elec-trostatic potential inside the dome. Since the potential term is much smallerthan one, this reduces to ∆ ν/ν = − Gϕ / k e c . (50)For an electrostatic accelerator of several M eV , ∆ ν/ν ∼ − , and thenull results in the experiments [18, 19] are expected, as proclaimed by theauthors. The conceptual error in this treatment is that the Reissner -Nordstr¨om metric is a solution to the Einstein - Maxwell field equationsfor an electrically charged spherical body of mass M and charge Q . Thethird term on the right of Eq.(48) is not an electrostatic term, rather it isa gravitational one just like the second term. The meaning of the thirdterm is as follows: The electric field outside the central body due to itscharge Q has an energy U field ∼ r ǫ | E | / M field = U field /c ∼ k e Q /c r . This mass gives rise to a gravitational poten-tial Φ g = GM field /r whose contribution to g must be proportional to Φ g /c which is approximately equal to Gk e Q /c R for r = R . This is the third15erm in g in Eq.(48). Thus on the surface of the dome g can be writtenas g = 1 − GM eff ( R ) /Rc . (51)with M eff ( r ) being the effective mass of the dome, the central body, givenby M eff ( r ) = M − k e Q rc , (52)where r ≥ R is the radial coordinate. Therefore, according to the meaning ofthe Reissner - Nordstr¨om metric, light moving in this spacetime is moving in agravitational field created by the mass M eff , and the redshift it may undergois a gravitational one. Certainly, it is incorrect to call the contribution of the Q - term an electrostatic redshift. In the present work, we have shown, employing classical electromagnetic the-ory, that positively charged objects age faster at high potential points thanthose at low potential points in an electric field. The aging of the negativelycharged objects, however, takes place in the opposite way. This electrostatictime dilation effect is similar to its gravitational analogue and there exists ageneral relativistic theory that predicts its existence [21]. We have demon-strated that this effect can also be obtained from an electrical equivalenceprinciple whose salient features have been elucidated. A Pound - Rebka -Snider experiment subjecting the source to a horizontal electric field wouldshow no electrical redshift because electric charge is conserved and is in-dependent of any forms of energy (which is not the case for mass). Thisis a result confirmed by two experiments that employed interferometers toobserve the questioned electrostatic redshift. If, on the other hand, it werepossible to measure the frequency change of photons in the gravitational fieldof the Earth with interferometers, one would observe no gravitational red-shift either. This is because the energy/frequency of photons do not changeas they move in a gravitational field, as discussed in [17]. An attempt in theliterature to explain the null results of the two electrostatic redshift exper-iments by making use of the Reissner - Nordstr¨om metric has been shownto be invalid. We cannot overemphasize the performance of the proposedexperiment with ionic clocks in a static electric field to confirm the electro-static time dilation effect presented in this work. Such a confirmation wouldalso support the unified theory of gravitation and electromagnetism leadingto the metric in Eq.(26). 16 cknowledgement
We thank C. J. de Matos of Institute of Aerospace Engineering, Technis-che Universit¨at Dresden, Germany for invaluable discussions and bringingreferences [9], [10], and [11] to our attention.
References [1] A. Einstein, Jahrb. Radioakt. Elektronik (1907) 411.[2] For an English translation of [1] see, H. M. Schwartz, Am. Jour. Phys. (1977) 899.[3] R. V. Pound, G. A. Rebka, Phys. Rev. Lett. (1960) 337.[4] R. V. Pound, J. L. Snider, Phys. Rev. (1965) B788-B804.[5] J. Hafele, R. Keating, Science (1972) 166.[6] C. Alley et al., in Experimental Gravitation , Proceedings of the Confer-ence at Pavia (Sept. 1976), ed. B. Bertotti (Academic Press, New York,NY, 1977).[7] R. F. C. Vessot, M. W. Levine, Gen. Rel. Grav., (1979) 181.[8] R. F. C. Vessot, et al., Phys. Rev. Lett (1980) 2081.[9] P. Delva, et al., Phys. Rev. Lett (2018) 231101.[10] S. Herrmann, et al., Phys. Rev. Lett. (2018) 231102.[11] C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland, Science (2010) 1630.[12] A. Schild, Texas Quarterly, (1960) 42.[13] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H.Freeman and Company, 1973.[14] R. P. Feynman, F. B. Morino, W. G. Wagner, Ed. by B. Hatfield, Feyn-man Lectures on Gravitation,, Addison - Wesley Publishing Company,1995.[15] B. F. Schutz, A First Course in General Relativity, Cambridge Univer-sity Press, 1999. 1716] J. L. Synge, Relativity: The General Theory, North - Holland PublishingCompany, Amsterdam, 1960.[17] L. B. Okun, K. G. Selivanov, and V. I. Telegdi, Am. J. Phys. (2000)115.[18] R. J. Kennedy, E. M. Thorndike, Proc. N. A. S. (1931) 620.[19] H. T. Drill, Phys. Rev. (1939) 184.[20] J. F. Woodward, R. J. Crowley, Nature (1973) 41.[21] M. ¨Ozer, Preprint at http//arXiv.org/gr-qc/9910062 (1999)[22] C. J. de Matos, M. ¨Ozer, and G. L. Izworski Preprint athttp//arXiv.org.1712.04347 (2017).[23] J. B. Hartle, Gravity, An Introduction to Einstein’s General Relativity,Addison Wesley, 2003.[24] S. Weinberg, Gravitation And Cosmology: Principles And ApplicationsOf The General Theory Of Relativity, (John Wiley & Sons, Inc., 1972).[25] R. C. Tolman, Relativity, Thermodynamics and Cosmology, Dover Pub-lications, Inc., 1987.[26] T. Rosenband et al., Science (2008) 1808.[27] C. W. Chou. D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, T. Rosen-band, Phys. Rev. Lett., (2010) 070802.[28] S. M. Brewer et. al., Phys. Rev. Lett., (2019) 033201.[29] It should be noted that, usually time dilation , frequency change of a clock , and redshift are all referred to as redshift in the literature. We will use a precise language and distinguish amongthem.[30] It should be noted that ∆ T ( H ) or ∆ T ∗ ( H ) are related to the oscillationfrequencies as ∆ T ( H ) / ∆ T (0) = f ( H ) /f (0), but not as ∆ T ( H ) / ∆ T (0) = f (0) /f ( H ) as one may expect. This is because the ticks of clocks at height0 may be normalized so that the time intervel between two ticks is onesecond. Then clocks at height H register a time interval of N ( H ) secondsfor N ( H )+1 ticks while those at height 0 register N (0) seconds for N (0)+1ticks. Obviously ( N ( H ) + 1) / ( N (0) + 1) = f ( H ) /f (0).1831] This is because, had it lost energy in moving up in the gravitationalfield, the photon of energy E at level 0 would have reached height H withan energy E − E gH/c . Hence the required Doppler speed would have beentwice as that in Eq.(12) so that the energy of the photon is E + E gH/c ,neglecting the c − and smaller terms.[32] This should not be confused with the Stark Effect which takes place in anelectric field as a result of the interaction of the field with the electric dipolemoment of the atom. The effect considered here takes place for electricallycharged systems with zero elecric dipole moment too, such as a proton, forexample. Furthermore, two charged atoms with the same electric dipolemoment in a uniform electric field but at different potentials would havethe same Stark effect energy level splittings whereas their aging would bedifferent due to the different electric potentials they are exposed to.[33] Naively thinking, one may be tempted to conclude from this expres-sion that the photons emitted at the bottom with frequency ν B reach thetop with frequency ν T such that ν T ≈ ν B (cid:16) − qm i | E | dc (cid:17)(cid:17)