# Mass or Energy: On Charge of Gravity

Authors: Zhong-Yue Wang
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The gravitational charge should be the energy instead of the mass. This modification will lead to some different results, and the experiments to test the new idea are also presented. In particular, we figure out how to achieve the negative energy and repulsive gravitational force in the lab.


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## Body

## 1. Introduction

A gauge theory requires the conserved charge. The massm0 is an invariant in relativity and some alternative theories where c2 is replaced by another constant K [1–4] but not conserved in the creation, annihilation, etc. Consequently, it is impossible to get the charge of gravitation. Indeed, a photon in free space can be pulled towards the star and Earth [5] although m0 is zero. In spite of momentum conservation, the momentum p is not the gravitational charge either because the stationary objects in Cavendish’s torsion-balance experiment can attract each other. We tend to regard the gravitational interaction as arising from conservation of energy E and predict some novel effects which cannot be explained by traditional theories.


## 2. New Form

Newton’s law of universal gravitation states that every mass attracts any other mass by a force. It takes the form(1)Gm1m2r2,where G=6.674×10−11 Nm2 kg−2 is the gravitational constant and r is the distance. If the charge of the gravitational interaction is energy, the new form of the force should be
(2)G′E1E2r2,and the potential
(3)φ=−G′Er,is a dimensionless quantity to indicate the deviation from the flat space-time in general relativity (Equation (49)). The Poisson equation is replaced by
(4)∇2φ=4πG′ℵ,where ℵ is the energy density of the source. It is reduced to
(5)∇2φc2=4πGρ,once the relation between ℵ and the mass density ρ of the source is
(6)ℵ=ρc2.At a low speed (v≪c),
(7)E=m0c21−v2/c2≈m0c2,G′E1E2r2≈G′m01m02r2c4.In comparison to Equation (1), the new gravitational constant G′ is
(8)G′=Gc4=8.262×10−45N−1.For instance, the energies of the Earth and a massless photon areE=Mc2≈M0c2 and hf (Planck constant h times the frequency f), respectively. The gravitational force
(9)G′Mc2r2hf≈G′M0c2r2hf=GM0r2hfc2.is nonzero. Equation (2) can be rewritten as
(10)G′E1E2r2=GE1/c2E2/c2r2.Here,E/c2 plays the role of the so-called gravitational mass mG. From now on, the concept mG is redundant, and the physical meaning of the principle of equivalence mi=mG is just m=E/c2. Einstein claimed that “The calling force of the earth depends on the gravitational mass. The answering motion of the stone depends on the inertial mass.” [6]. It should be revised to “The calling force of the earth depends on the energy. The answering motion of the stone depends on the mass”.


## 3. Negative Energy and Repulsion

In Newton’s theory, the gravitational force is always attractive. Now, we use the new form to examine a bound system. The rest energy of a deuteron is1875.6×106eV, and the force between the Earth should be
(11)G′M0c2r21875⋅61×106eV=GM0r21875⋅61×106eVc2.Nevertheless, the deuteron is composed of one proton and one neutron. Their rest energies are938.27×106 eV and 939.57×106 eV. The resultant of forces
(12)G′M0c2r2938.27×106eV+G′M0c2r2939.57×106eV=G′M0c2r21877.84×106eV,is larger than (11). Actually, the gravitational force between the negative binding energy −2.23×106 eV and the Earth should be repulsive and Equation (11) is equal to
(13)G′M0c2r2938.27×106eV+G′M0c2r2939.57×106eV−G′M0c2r22.23×106eV.Like the Coulomb force, gravity can be not only pulling but also repelling.


## 4. Gravitational Effect of a Potential Energy

The total energy in the above example is still positive. Let us consider an object whose total energy can be negative. The wave function of a free particle is(14)ψ~expiℏEt−px=expiℏmc2t−px,m=m01−v2/c2.In a Faraday cage, the electrostatic fieldE is absent even though an electric scalar potential φE is applied. In practice, φE is usually the voltage relative to ground. The velocity v or momentum p of a particle electrically charged q does not change while the total energy is
(15)E=mc2+qφE.Generally speaking, energy is related to the momentum, and the energy shift is accompanied by the change of momentum. However, this is a special state whose momentum and velocity remain unchanged as the electrostatic field strength is zero. The feature is decisive to the success of the experiment to detect an effect caused by the force of gravity which is much weaker than other forces. The wave function is now(16)ψ~expiℏEt−px=expiℏmc2t+qφEt−px,and the particle gains an extra phase [7]
(17)qℏφEt.It is the evidence of Equation (15). In classical mechanics, the gravitational force between the Earth is
(18)GM0r2m0≈m0g,g≈GM0r2=9.8ms−2.Using the new law (Figure1),
(19)G′M0c2r2mc2+qφE=GM0r2m+qφEc2=GM0mr21+qφEmc2=mg1+qφEmc2,(20)g′=G′M0c2/r2mc2+qφEm=GM0/r2m+qφE/c2m=g1+qφEmc2.Figure 1
Gravitational accelerations.It is against common sense that the gravitational acceleration of a freely falling body is independent of the mass, which has lodged itself in the public mind since the anecdotal Galileo’s Leaning Tower of Pisa experiment. We should measure the gravitational accelerations of electrically charged particles [8, 9] in a region where φE≠0 and ∇φE=0. For example, the electric charge of an electron is q=−e and the force
(21)mg1−eφEmc2,can be zero on condition that
(22)φE=mc2e.The critical potential of a slow electron is(23)mc2e≈m0c2e=0.51×106volts.It must be said that we get a negative total energy (15), repulsive force of gravity (21), and reversed acceleration g′ (20) of an electron if
(24)φE>mc2e≈0.51×106volts.


## 5. Influence on the Mass

A hypothesis to avoid a nonconstant acceleration (Equation (20)) is that m is changed to
(25)m⟶m+qφEc2,simultaneously. Under the circumstances, the gravitational acceleration is
(26)G′M0c2/r2mc2+qφEm+qφE/c2=GM0/r2m+qφE/c2m+qφE/c2=g=9.8ms−2,as before, and the gravitational interaction is still equivalent to a “geometric effect.” It is difficult to test Equation (20) directly in normal labs [8, 9], and one can reexamine the physical quantities involving m or m0 to speculate on the gravitational acceleration. For instance, the specific heat Cv of the electron gas is proportional to the temperature T and m0, i.e.,
(27)Cv=γT∝T,γ∝m0.In a Faraday cage, the specific heat will be(28)Cv∝m0−eφEc2T, in the event that
(29)m0⟶m0−eφEc2.Now, we discuss the spectra emitted by hydrogen atoms in a cage. The electric potential energy of an electron in this atom is(30)−e24πε0R−eφERisthedistancetotheproton.The electric force as the gradient of Equation (30) is still
(31)e24πε0R2.Hence,(32)e24πε0R2=mv2R.Due to Bohr’s quantization condition,(33)mvR=nℏn=1,2,3⋯,the total energy of an electron is
(34)E=mc2−e24πε0R−eφE≈m0c2−e432π2ε02ℏ2n2m0−eφE, whereby the frequency of the spectrum should be
(35)hf=E1−E2=e432π2ε02ℏ2m01n22−1n12=13.6eV1n22−1n12.In my opinion,m or m0 is unaffected; otherwise, we shall discover a new effect in spectroscopy
(36)hf⟶e432π2ε02ℏ2m01−eφEm0c21n22−1n12=13.6eV1−eφEm0c21n22−1n12.


## 6. Superconducting Interferometry Gravimeter

The electrostatic field within a superconductor vanishes as well. Inspired by the COW experiment of the neutron [10], we design a superconducting circuit (Figure 2) to detect the phase shift caused by the weight of the carrier.Figure 2
Schematic diagram.At point 1, the incident supercurrent is split into two parts on a horizontal plane1234. They follow the path 124¯ and 134¯, and the relative phase at point 4 where they recombine is
(37)p0ℏL+H−p0ℏH+L=0.p0=mv0≈m0v0 is the initial momentum. By rotating the interferometer about the line 12¯, the difference between the lower path 124′¯ and upper path 13′4′¯ is
(38)ϑ=p0−pℏL.The height of3′4′¯ is H and the momentum p should be
(39)p22m0=p022m0−mGgH.In the COW experiment,(40)p022m0≫mGgH,p≈p0−2m0mGgHp0=p0−2mGgHv0.The phase shift(41)ϑ=p0−pℏL=2mGgHℏv0L∝mGg,is proportional to the gravitational force mGg≈m0g. The experiments suggested in Sections 4–5 are to determine the gravitational acceleration and mass, respectively. This proposal is to weigh a superconducting carrier. It should be multiplied by a factor
(42)m0g⟶m0g1+qφEm0c2after an electric scalar potential φE is applied. In Einstein's elevator, the inertial force is inadequate to compensate for the gravitational force (Equation (42)) and the phase shift is nonzero.


## 7. Physical Significance

The gravitational acceleration (20) is at variance with not only Newton’s theory but also Einstein’s general relativity whose motion equation
(43)d2xμds2+Γαβμdxαdsdxβds=0,is independent of the mass. Actually, a geometric description holds true for the constant gravitational charge-to-mass ratios. In relativity, it is mG/mi=1. We point out that the gravitational charge is E and mG/mi=1 is equivalent to E/m=c2. However, the ratio in the above counterexample
(44)Em=mc2+qφEm=c2+qφEm,is not constant.


## 8. Geometric Theories of Gravity

To a constantK, there are [1–4]
(45)ds2=gμνdxμdxν,(46)dxμdxν=Kt2+dx2+dy2+dz2,(47)gμν=−1000010000100001,to describe an inertial reference frame. In a gravitational field, Equations (45) and (46) are still tenable, and gμν is given by Einstein’s field equation
(48)Rμν−12gμνR=8πG′TμνG′=8.262×10−45N−1.For the sake of convenience, we consider the simplest case(49)gμν=−1+2φ000011+2φ0000r20000r2sin2θ.In view ofE≈M0c2 of the source, another expression of Equation (49)
(50)gμν=−1−2GM0c2r000011−2GM0/c2r0000r20000r2sin2θ,is familiar to us, and Equation (45) can be written as
(51)ds2=−1−2GM0c2rKdt2+dr21−2GM0/c2r+r2dθ2+r2sin2θdϕ2.SupposeK=c2, it is the well-known Schwarzschild solution
(52)ds2=−1−2GM0c2rc2dt2+dr21−2GM0/c2r+r2dθ2+r2sin2θdϕ2.As to the gravitational field produced by the electrically charged particle in Section4, there is an extra term 2G′qφE/r=2GqφE/c4r in the Reissner-Nordström solution.The approximation of the motion equation is(53)d2xiKdt2+∇φ=0,and the acceleration should be
(54)a=d2xidt2=−K∇φ.On the Earth,(55)φ=−G′Er=−Gc4M0c2r=−GM0c2r,(56)a=Kc2GM0r2.In the age of Newton, the energy-mass equations of all experimental objects satisfyK=c2. Thus,
(57)a=GM0r2.This is just Newton’s law(58)m0a=GM0r2m0.K≠c2 is conducive to construct MOND (modified Newtonian dynamics).


## 9. Negative Mass and Attraction

A negative mass was inconceivable in Newton’s time, whereas scientists can make anomalous waves in metamaterials now whose wave vectors are reversed. The phenomena imply that the masses of quanta of these waves are less than zero [4]. In the light of Newton’s formula (Equation (1)), the gravitational force between the quanta and Earth should be repulsive. However, the energy hf of such a quantum is positive and the force
(59)G′M0c2r2hf,is still attractive. The sign of gravity depends on the product of energies rather than masses. Interestingly, the gravitational acceleration
(60)G′M0c2/r2hfmm<0,should be in the opposite direction. From another angle, it is because K<0 [4] in Equation (56). In Section 4, the force is repulsive and mass is positive. Conversely, here are the attractive force and negative mass. Both the accelerations turn towards outer space. They are two types of antigravity propulsion.


## 10. Metric Tensor and Noninertial Effect

In a rotating frame,(61)ds2=−1−Ω2r2KKdt2+dr2+r2dθ2+dz2+2Ωr2dθdt.Ω is the angular frequency of the rotation. The relation between frequencies f1 and f2 at different distances should be
(62)f11−Ω2r12K=f21−Ω2r22K.WhenK=c2,
(63)ds2=−1−Ω2r2c2c2dt2+dr2+r2dθ2+dz2+2Ωr2dθdt,(64)f11−Ω2r12c2=f21−Ω2r22c2.It was verified long ago [11]. In fact, there are following similarities between the photon and phonon (quantum of sound) (Table 1):Table 1PhotonPhononSpeedc=1/ε0μ0 (constant)Cs=modulus/density (constant)EnergyE=ℏωE=ℏωMomentump=ℏkp=ℏkEnergy-momentum relationE=pc (linear)E=pCs (linear)Massm0=0 (massless)m0=0 (massless)Space and time are not physical realities. They are tools to reflect nature, and one can attempt different space-time structures to fit the data. For example, the coefficientK in Equation (46) should be Cs2 to describe the flat space-time
(65)ds2=−Cs2dt2+dr2+r2dθ2+r2sin2θdϕ2,of sound [1]. In a rotating system,
(66)ds2=−1−Ω2r2Cs2Cs2dt2+dr2+r2dθ2+dz2+2Ωr2dθdt,(67)f11−Ω2r12Cs2=f21−Ω2r22Cs2.Neither classical mechanics nor relativity whereK is fixed as c2 and the factor is 1−v2/c2≪1−v2/Cs2 can interpret Equation (67) which was predicted in 2000 [1]. Nonetheless, it was observed in 2011 [12]. K=c2 of light is much greater than K=Cs2 of sound, so the noninertial shift of light [11] is much less than that of sound [12].


## 11. Comparison between the Gravitational Field and Noninertial Frame

K=Cs2 in Equation (51) yields the equations of sound in a gravitational field
(68)ds2=−1−2GM0c2rCs2dt2+dr21−2GM0/c2r+r2dθ2+r2sin2θdϕ2.(69)f11−2GM0c2r1=f21−2GM0c2r2,The gravitational frequency shift is as small as that of light. The result can be derived from a nongeometric theory too. Equation (9) is valid no matter whether it is a photon or phonon, and the potential energy should be
(70)−G′M0c2rhf=−GM0rhfc2.In this sense, the photon and phonon behave as if they have the same “gravitational mass”hf/c2, though the inertial mass of the latter is hf/Cs2 [1, 12]. The total energy in a gravitational field should be
(71)hf−GM0rhfc2.It is conserved(72)hf1−GM0r1hf1c2=hf2−GM0r2hf2c2,f11−GM0c2r1=f21−GM0c2r2.We have the Mössbauer effect to measure the gravitational frequency shift of light [5] but no technologies to detect such a tiny change of sound so far. In contrast, the gravitational shift of sound ought to be observable by substituting the mass hf/Cs2 of a phonon determined by the noninertial experiment [12] into Newton’s law. The potential energy and total energy near the surface of the Earth are
(73)−GM0rhfCs2,(74)hf−GM0rhfCs2=hf1−GM0Cs2r.Namely,(75)f11−GM0Cs2r1=f21−GM0Cs2r2.Nevertheless, there is no need to test Equation (75) experimentally because it does not agree with the acoustooptic effect [13]. An incident photon hfi can absorb the energy hf of a phonon, and the relation between the diffracted photon hfd is
(76)hfi+hf=hfd,(77)fi+f=fd.Equation (76) does not allow for the gravitational interaction. According to Equation (71) of the photon and Equation (74) of the phonon, their energies in this process are
(78)hfi1−GM0c2r+hf1−GM0Cs2r=hfd1−GM0c2r.Owing toGM0/c2r≪1,
(79)hfi+hf1−GM0Cs2r=hfd.A typical speed in the acoustooptic material isCs=5000ms−1, and the total energy of a phonon is
(80)hf1−GM0Cs2r≈hf1−grCs2=hf1−9.8ms−2×6.4×106m50002m2s−2≈−1.5hf.Therefore, Equation (79) is
(81)hfi−1.5hf=hfd,fi−1.5f=fd.It is inconsistent with the experimental fact (77). We have to conclude that both the photon and phonon are subject to Equation (71) and the law of energy conservation in a gravitational field is
(82)hfi1−GM0c2r+hf1−GM0c2r=hfd1−GM0c2r.In a geometric theory, it is(83)hfi1−2GM0c2r+hf1−2GM0c2r=hfd1−2GM0c2r.The gravitational shifts of light and sound are the same, but their noninertial shifts (Equations (64) and (67)) are unequal. That is to say, in a geometric description, gμν of the gravitational field only depends on the source and has nothing to do with K of the test particle while gμν of a noninertial frame is associated with not only the acceleration but also K. A gravitational field cannot be equated with the noninertial system, unless K=c2.


## 12. Conclusions

Newton’s law of universal gravitation is not universal. The charge of gravity should be the energy whose concept became mature in the 19th century, about 100 years after his death. For this reason, the electromagnetic radiation and neutrinos in the cosmos participate in the gravitational interaction no matter if they are massive or not. In general, the mass-energy equation of common objects isE/m=c2, whereby Newton’s law is applicable to most cases. Likewise, Einstein’s general relativity is effective under the same premise of E/m=c2. We came up with some exceptions which can be divided into two types. One is E/m=constant≠c2 [1–4] corresponding to a geometric description (Sections 8–11). The other is E/m≠constant [14, 15], and this paper proposes a new counterexample that the energy as the gravitational charge is changed by the potential (Equation (15)) which has no effect on the mass (Section 5) and the gravitational charge-to-mass ratio is no longer c2 (Equation (44)). We hope the experiments in Sections 4–6 can be carried out as soon as possible.



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