Force, curvature, or mass: disambiguating causes of uniform gravity
FForce, curvature, or mass: disambiguating causes of uniform gravity
Yuan Shi ∗ Lawrence Livermore National Laboratory, Livermore, California 94551, USA (Dated: August 7, 2019)In addition to Newtonian forces and spacetime curvature, gradients of the Higgs vacuum expecta-tion value (VEV), which can be induced by the presence of matter, also lead to particle accelerationand photon redshift. Here, I compare distinct effects of force, curvature, and Higgs VEV gradientthat cause uniform acceleration. In particular, I show that a spurious stress-energy tensor is requiredif the acceleration is in fact due to the Higgs VEV gradient but is falsely attributed to spacetimecurvature. On cosmological scales, the spurious density coincides with the observed dark energydensity and may contribute to the Hubble tension; on galactic scales, the inferred dark matter den-sity falls within expectation and may explain the lopsidedness of galaxy spectrographs; and on theEarth scale, the spurious density is minuscule. The experimental precision required to disambiguatecauses of the Earth’s gravity is estimated. Laboratory tests are challenging but possible.
Einstein’s theory of general relativity (GR) interpretsgravity as spacetime curvature [1–3]. Constructed to re-cover Newtonian gravity in the weak-field limit, GR isinitially supported by three additional evidences: grav-itational redshift of photons [4–8], perihelion processionof Mercury [9–13], and gravitational lensing of light [14–19]. After almost a century, other predictions of GRhave finally been confirmed, owing to recent detectionof gravitational waves [20, 21] and an image of a blackhole [22]. The classical GR would be complete if it werenot due to outstanding difficulties on large scales [23–25],which have motivated alternative theories [26–28]. Mostnotably, the standard cosmology model requires ∼ ∼
27% dark matter [29–32], which haveso far evaded laboratory detection [33].While the geometry that enters the Einstein’s equationis classical, matter that curves the spacetime is quantum.At low energy, quantum matter gravitates because of itsmass, and in the Standard Model, elementary particlesacquire their masses through the Higgs mechanism [34].Although nucleons, which dominate the mass of ordinarymatter, are composite particles whose masses are moresubtle [35–38], dimensionful scales are ultimately set byelementary particles. For example, the scale can be setby the pion decay constant, which is related, throughelectroweak processes, to the mass of W bosons.It is important to recognize that the Higgs VEV, andtherefore the masses of all particles, can be altered byambient matter. The simplest argument can perhaps bemade via the toy Lagrangian ( (cid:126) = c = 1 unless specified): L = ¯ ψi /∂ψ − f φ ¯ ψψ + 12 ( ∂φ ) + 12 u φ − λ φ , (1)where the Dirac fermion ψ couples with the real scalar φ via the Yukawa coupling f . Once φ acquires a VEV,the fermion gains a mass term m = f (cid:104) φ (cid:105) . Moreover, theVEV is depleted by the presence of a nonperturbativeamount of matter that dominates antimatter: (cid:104) φ (cid:105) = 2 (cid:114) u λ cos π − θ (cid:39) (cid:114) u λ − f (cid:104) ¯ ψψ (cid:105) u , (2) FIG. 1. Uniform gravity due to force (a), frame accelera-tion (b), and Higgs VEV gradient (c) can be disambiguatedusing a second test particle (red, “1”) with different initialconditions, in addition to a prototypical particle (grey, “0”)that has constant a . For Newtonian forces (a), a = a . Ingeneral relativity (b), a is usually not a constant. For massgradient (c), a usually differs from a but is a constant. where cos θ = 3 (cid:112) λ/ f (cid:104) ¯ ψψ (cid:105) / (2 u ). The VEV given byEq. (2) is not the global minimum, but it gives a positivefermion mass and is stable as long as cos θ <
1. Nearthis local minimum, the Higgs field acquires a mass M = u [4 cos ( π − θ ) − (cid:39) u − (cid:112) λ/ f (cid:104) ¯ ψψ (cid:105) /u . The Higgsmass vanishes when θ = 0, at which the local minimum islost. There, the density ρ = f (cid:104) φ (cid:105)(cid:104) ¯ ψψ (cid:105) attains the criticalvalue ρ c = 4 u / (3 λ ) = (cid:104) φ (cid:105) M /
9, where “0” denotes thevacuum case. For the Higgs field [33], (cid:104) φ (cid:105) ≈
246 GeVand M ≈
125 GeV, so ρ c ≈ . × kg · m − . We seematter effects are small even at nuclear density. Beyondthis toy model, φ is an SU(2) doublet, and (cid:104) ¯ ψψ (cid:105) hasspacetime dependencies. Nevertheless, the key ingredientof the argument remains effective.A spacetime-dependent Higgs VEV is analogous to adensity gradient in refractive media (Fig. 1). As firstnoted by Anderson [39, 40], photons become massiveparticles in plasmas, whose dispersion relation ω = ω p + c k allows one to identify the plasma frequency ω p = (cid:112) e n /(cid:15) m e as the photon mass. When the plasmadensity n is nonuniform, refraction occurs, which mim-ics effects of gravity [41]. However, if one is unaware ofthe plasma, then the bending of light rays would requiresome spurious forces or spacetime curvature to explain. a r X i v : . [ g r- q c ] A ug To quantify the effects, let us focus on a classical par-ticle, namely, the geometric-optics approximation of aquantum wave packet. The classical trajectory x µ ( τ ) ona prescribed background field is described by the action (cid:90) dτ (cid:16) m ˙ x µ g µν ˙ x ν + e ˙ x µ A µ (cid:17) , (3)where g µν is the metric tensor and A µ is the electromag-netic gauge 1-form. Notice that while g µν and m can havespacetime dependencies, the charge e must be a constantdue to gauge symmetry.The geodesics, or classical trajectories, are obtained byextremizing the action. The resultant equation is¨ x α + (Γ αµν + S αµν ) ˙ x µ ˙ x ν = em g αµ F µν ˙ x ν , (4)where Γ αµν is the usual Christoffel symbol and F µν is theelectromagnetic tensor. The extra term due to the HiggsVEV gradient is S αµν = 12 (cid:104) φ (cid:105) (cid:16) δ αµ (cid:104) φ (cid:105) ,ν + δ αν (cid:104) φ (cid:105) ,µ − g µν g ασ (cid:104) φ (cid:105) ,σ (cid:17) , (5)where commas denote partial derivatives. Since F µν = − F νµ , Eq. (4) implies that H = m ˙ x µ g µν ˙ x ν is a constant.That the Yukawa coupling does not enter has the im-portant implication that S αµν is universal. In other words,the motion of all massive particles are equally affectedby Higgs VEV gradients. On the other hand, light-likegeodesics are not affected by the conformally flat S αµν .This is intuitive, because photons have no direct cou-pling with the Higgs field. Mathematically, after Weylrescaling [42], a spacetime-dependent mass may be ab-sorbed by redefining the metric to be ( mg ) µν . Then,( mg ) µν = g µν /m , and the Christoffel symbol Γ αµν [ mg ] =Γ αµν [ g ] + S αµν . However, the role played by g µν , which isdimensionless, is physically distinct from the role playedby the Higgs VEV, which is dimensionful. In fact, (cid:104) φ (cid:105) plays a similar role as the scalar gravity of Nordstr¨om[43], which predates GR and is later incorporated intothe gravitation theory of Ni and Yang [44, 45].A spacetime-dependent mass mimics effects of a curvedmetric. Quantitatively, the metric connection definedby ( mg ) µν gives rise to a Riemann curvature tensor R µναβ [ mg ] = R µναβ [ g ] + C µναβ + r µναβ , where C µναβ =Γ µασ S σνβ + S µασ Γ σνβ − ( α ↔ β ) and r µναβ = S µνβ,α + S µασ S σνβ − ( α ↔ β ). What is of importance to GR is the Ricci cur-vature tensor, to which the Higgs VEV contributes r µν = 3 (cid:104) φ (cid:105) ,µ (cid:104) φ (cid:105) ,ν (cid:104) φ (cid:105) − (cid:104) φ (cid:105) ,µν (cid:104) φ (cid:105) − (cid:104) φ (cid:105) ∂ α (cid:16) g µν g αβ (cid:104) φ (cid:105) ,β (cid:17) . (6)Notice that r µν can be nonzero even when g µν is trivial.In other words, this effective curvature has little to dowith the intrinsic geometry of spacetime, and it arisesonly because one tries to interpret the Higgs VEV usinga geometric language. To illustrate the central idea, let us consider the con-ceptually important, albeit not realistic, case of uniformacceleration. A particle is undergoing uniform accelera-tion if its acceleration is a constant in the particle’s in-stantaneous rest frame [46]. When observed in the labframe, uniform acceleration translates to the condition γ d x /dt = a , where γ is the Lorentz factor and a is aconstant. Suppose a is along the z axis, then by sym-metry, there exist a coordinate system, which I will re-fer to as the lab frame, where the action is of the form (cid:82) dτ [ m ( U ˙ t − V ˙ z − ˙ x − ˙ y ) − eE z ˙ t ], where m , U , and V only depends on z , and E is a constant electric field,whose effect is identical to that of a Newtonian gravita-tional field. Then, the equation of motion (EOM) in x , y ,and t directions can be easily integrated to give m ˙ x = p x , m ˙ y = p y , and mU ˙ t − eE z = p t , where p x , p y , and p t are three additional constants of motion. Substitutingthese into the EOM in the z direction, (cid:16) dzdt (cid:17) = U V (cid:104) − (cid:16) Up t + eE z (cid:17) ( Hm + p x + p y ) (cid:105) . (7)This equation locally determines a unique geodesics fora given set of initial conditions. In what follows, I willfocus on the special case where the motion is along the z direction, for which p x = p y = 0. The initial conditionsat τ = 0 are t = 0, z = z , ˙ t = γ , and ˙ z = γ β . Then,the nontrivial constants of motion are p t = m γ U − eE z and H = m γ ( U − β V ), where m denotes m ( z ) evaluated at z and so on.Before discussing the general case where force, curva-ture, and mass are simultaneously at play, it is helpfulto consider their effects individually. First, in the caseof uniform force, U = V = 1 and m = m is a constant(Fig. 2, dotted). For time-like geodesics, γ (1 − β ) = 1,and Eq. (7) becomes ( dz/dt ) = 1 − / [ γ + a E ( z − z )] .The trajectory is hyperbolic:[ a E ( z − z ) + γ ] − ( a E t + γ β ) = 1 , (8)where a E = eE /m is the bare acceleration due to aconstant force field.Second, let us consider uniform gravity in GR, which isindistinguishable from a uniform frame acceleration ac-cording to the equivalence principle. In this case, E = 0, m = m , and the spacetime is trivial. Nonzero compo-nents of the Christoffel symbol are Γ ttz = U (cid:48) /U, Γ zzz = V (cid:48) /V , and Γ ztt = U U (cid:48) /V , where the prime denotes thederivative with respect to z . Components of the Rie-mann curvature tensor are either zero or proportional to R tzzt = U (cid:48)(cid:48) /U − U (cid:48) V (cid:48) / ( U V ). For uniform gravity, thespacetime is Ricci flat, for which V ∝ U (cid:48) . As is wellnoted by Rohrlich [47], this condition does not uniquelydetermine the metric.The metric can nevertheless be fixed by imposing coor-dinate conditions. A unique metric exist, which satisfiesboth conditions (i) the trajectory of a prototypical testparticle with z = 0 and γ = γ is hyperbolic, and (ii)the metric asymptotes to U (cid:39) − a G z and V (cid:39) z →
0. Here, a G is the uniform frame acceleration, andI have used the sign convention that the gravitationalpotential is lower for larger z . This unique metric is U = cosh w cosh ¯ w , V = cosh w tanh w sinh ¯ w cosh ¯ w , (9)where cosh w = γ , ¯ w = sinh w − sinh w + w , andcosh w = a G z +cosh w . This change of variable is invert-ible only when ¯ w > w − sinh w and a G z > − cosh w .The metric can be trivialized by the coordinate trans-formation ( t, z ) → (˜ t, ˜ z ) with a G ˜ t = U ( z ) sinh( a G t + w ) − sinh w , (10) a G ˜ z = cosh w − U ( z ) cosh( a G t + w ) . (11)The metric tensor ˜ g then becomes the Minkowski metric.Moreover, the trajectory of the prototypical test particleis ˜ z = 0. In other words, the tilde coordinate defines thefree-fall frame of the prototypical test particle. Noticethat Eqs. (10) and (11) become a Lorentz boost when a G →
0. On the other hand, when a G is nonzero, thecoordinate transform does not cover the entire spacetime.For the lab origin, the Rindler horizon is located at a G (˜ z + c ˜ t ) = cosh w − sinh w . (12)The accelerating lab observer will never see any particlecrosses the horizon nor receive any photon beyond thehorizon, even though the spacetime is trivial.Given the flat metric, the geodesics (Fig. 2, solid) arestraight lines in the free-fall frame. In the lab frame,Eq. (7) becomes d ¯ w/dt = a G (cid:112) cosh ¯ w − (cid:15) / sinh ¯ w ,where (cid:15) = cosh ¯ w − β sinh ¯ w / tanh w is given byinitial conditions. When (cid:15) >
0, the trajectory iscosh ¯ w = √ (cid:15) cosh( a G t + θ ) , (13)where cosh θ = cosh ¯ w / √ (cid:15) . This is a time-likegeodesic ˜ z = ˜ z + ˜ β ˜ t , where a G ˜ z = (cosh ¯ w − cosh w ) / [ √ (cid:15) cosh( θ − w )] and ˜ β = tanh( θ − w ).The geodesics become light-like when (cid:15) = 0, and space-like when (cid:15) <
0, whose equations can be found similarly.Unlike the case of a constant force, uniform gravity inGR has a peculiar feature: the motion of a test particleis hyperbolic if and only if its initial conditions satisfy β = tanh w . Apart from this one-parameter family,other test particles do not experience a uniform acceler-ation, even through the gravity is presumably uniform.This peculiar feature can be traced back to Eqs. (10) and(11). On the other hand, any fixed point in the lab frameundergoes uniform acceleration in the free-fall frame, ex-cept that the acceleration a G /U depends on z .Third, having discussed well-established causes of uni-form acceleration, let us consider effects of the HiggsVEV gradient by setting U = V = 1 and E = 0. Then, Eq. (7) becomes ( dz/dt ) = 1 − m/ ( m γ ). For the pro-totypical test particle to undergo hyperbolic motion, (cid:104) φ (cid:105) = (cid:104) φ (cid:105) (cid:16) a M zγ (cid:17) − , (14)where a M is the bare acceleration due to the Higgs VEVgradient. As expected, (cid:104) φ (cid:105) is depleted in the directionof gravity. Although (cid:104) φ (cid:105) blows up at z = − γ /a M , itonly needs to take the form of Eq. (14) in the vicinity of z ∼
0, and the Higgs VEV can safely level off far abovethe ground without being noticed by the lab observer.Given the form of the Higgs VEV, the trajectories(Fig. 2, dashed) are always hyperbolic:( a M z + γ ) − ( a M t + κ γ β ) = κ , (15)where κ = ( a M z + γ ) /γ . This is similar to the case ofa uniform force, except that now the uniform acceleration a M = a M /κ depends on initial conditions.In addition to causing particle acceleration, a HiggsVEV gradient also causes gravitational redshift evenwhen the spacetime is trivial. In one scenario, we canmeasure the photon frequency using an atomic clock,whose rate is proportional to the Rydberg energy R y = m e e / (8 (cid:15) h ). Alternatively, we can measure the photonwavelength using a grating, whose size is proportional tothe Bohr radius a B = 4 π(cid:15) (cid:126) / ( m e e ). Since the massbecomes larger at higher gravitational potential, wherethe clock ticks faster and the ruler spans shorter, a pho-ton appears to change color when measured at differ-ent altitudes. Notice that d z/dt = −(cid:104) φ (cid:105) (cid:48) / (2 γ (cid:104) φ (cid:105) ), so a M equals to the redshift acceleration only for a one-parameter family of initial conditions.If the uniform acceleration in the form of Eq. (15) is at-tributed to spacetime curvature, then spurious dark mat-ter or dark energy is required in the Einstein’s equation.Denoting (cid:104) φ (cid:105) = (cid:104) φ (cid:105) e ζ , which is not necessarily caused FIG. 2. Example trajectories in lab frame (a) and free-fallframe (b) due to uniform force (dotted), frame acceleration(solid), and Higgs VEV gradient (dashed). For the prototyp-ical particle (blue), these causes are indistinguishable. How-ever, for a test particle with z = 0 but β = 5 β = 0 . β = β but z a/c = 3 (green), the trajectoriesare different. The dashed gray line is the Rindler horizon. by the presence of any matter, nonzero components of S αµν are S tzt = S xzx = S yzy = S ztt = S zzz = − S zxx = − S zyy = ζ (cid:48) /
2. Then, components of the effective Riemann tensorare either zero or proportional to r tzzt = r xzzx = r yzzy = ζ (cid:48)(cid:48) / r xtxt = r ytyt = r xyyx = ζ (cid:48) /
4. The effective Riccitensor is diagonal, with r tt = − r xx = − r yy = ( ζ (cid:48)(cid:48) + ζ (cid:48) ) / r zz = − ζ (cid:48)(cid:48) /
2. The effective Ricci scalar is then r = 3( ζ (cid:48)(cid:48) + ζ (cid:48) / r µν − rg µν / − Λ g µν = 8 πGt µν , where G is thegravitational constant and Λ is the bare value of thecosmological constant, then, the spurious stress-energytensor t µν takes the form of an ideal anisotropic fluid.The energy density ρ and the perpendicular pressure p ⊥ are ρ = − p ⊥ = ( (cid:104) φ (cid:105) (cid:48) / (cid:104) φ (cid:105) −(cid:104) φ (cid:105) (cid:48)(cid:48) / (cid:104) φ (cid:105)− Λ ) / (8 πG ), andthe parallel pressure is p (cid:107) = ( (cid:104) φ (cid:105) (cid:48) / (cid:104) φ (cid:105) +Λ ) / (8 πG ). Inthe special case of uniform gravity [Eq. (14)], t µν = ρg µν takes the vacuum form, where ρ = − [Λ + 3( a M ) / ( γ + a M z ) ] / (8 πG ) cannot be produced by ordinary matter.Depending on the value of Λ , the spurious fluid cor-responds to either dark energy or dark matter. More-over, when Λ < ρ ( z ) changes sign: the spurious fluidchanges from dark matter to dark energy along the di-rection of the gravity. The characteristic scale is ρ s = ( a M ) πGc , (16)where a M /c ∼ (cid:104) φ (cid:105) (cid:48) / (cid:104) φ (cid:105) is determined by the gradientscale length of the Higgs VEV, whereas (cid:104) φ (cid:105) may con-tribute separately to the vacuum energy density.On cosmological scales, the expansion of the universe isassociated with a M ∼ − cqH , where q ≈ − .
55 is the de-celeration parameter and H ≈
70 km · s − · Mpc − is theHubble parameter. Then, ρ s ∼ − kg · m − coincideswith the observed dark energy density. The key pointhere is that cosmological redshifts likely contain contri-butions from (cid:104) φ (cid:105) , which is more depleted by the highermatter density along the past light cone. This additionalredshift may contribute to the Hubble tension [48].For galaxies, if we take a M ∼ (150 km · s − ) / (10 kpc)to be the orbiting acceleration, then ρ s ∼ − kg · m − .To estimate an upper bound, take (cid:104) φ (cid:105) / (cid:104) φ (cid:105) (cid:48) ∼ ρ s ∼ − kg · m − . The observed dark matter densityfalls within the above estimates. The depletion of (cid:104) φ (cid:105) by matter in the galaxy creates a mass barrier that maycontribute to the confinement of gases and stars.Near the surface of Earth, the gravity is approxi-mately uniform. We can take a M = a ⊕ ≈ . · s − ,then ρ s ∼ − kg · m − , which is hardly noticeable incomparison to the density of ordinary matter.Next, let’s consider how to disambiguate causes of uni-form gravity when force, curvature, and mass all con-tribute. In this case, Eq. (7) can be integrated numer-ically, whereby a G and a M can be determined by fit-ting trajectories of test particles. For contained labora-tory tests on Earth, since az /c ∼ − ( z /
10 m) and β ∼ − ( v /
10 m · s − ) are minuscule, it is sufficientto Taylor expand the solution. To leading order in time, z (cid:39) z + v t + ( a E + a G + a M ) t + O ( t ), where a E = a E γ (cid:16) a M z γ (cid:17) ¯ C γ (cid:16) T ¯ T − β (cid:17) , (17) a G = a G (cid:104) T ¯ T + β (cid:16) S C − T ¯ T (cid:17)(cid:105) , (18) a M = a M γ + a M z (cid:16) T ¯ T − β (cid:17) . (19)Here, ¯ C := cosh ¯ w , T := tanh w , S := sinh w , andso on. For the above expansion, we only need Eq. (9)for g µν and Eq. (14) for (cid:104) φ (cid:105) near z ∼
0, keeping inmind that they are probably not good approximations ofthe Earth’s gravity. Notice that although the gravity ispresumably uniform, the accelerations have gradients.Unlike usual tests of the equivalence of free fall, wheredifferent test particles have comparable initial conditions[48–51], here it is crucial that their initial conditions aredistinct. In other words, although the free fall is universalfor all particles, their identical acceleration does dependon the initial height and velocity.In one scenario, we can launch particles with differ-ent β from z = 0. Denoting a = a G + a M /γ , theacceleration at β = 0, then the residual acceleration is a r (cid:39) (cid:0) a − b (cid:1) β − a β + . . . , (20)when we set a E = − a . Here, b = a G / [ γ (1 + γ )] is due to GR. When the uniform gravity is en-tirely due to GR (Fig. 3a, red), we have a = 2 b if γ = 1. Then, a r (cid:39) − . × − m · s − ( a G /a ⊕ ) v ,where v = v / (300 m · s − ). On the other hand, if theuniform gravity is entirely due to the Higgs VEV gradi-ent (Fig. 3a, blue), a r (cid:39) . × − m · s − ( a M /a ⊕ ) v .These two extreme cases scale differently with β and areof opposite signs.In another scenario, we can suspend test particles with β = 0 at different heights z . Suppose an electric field issetup to cancel the leading gradient of the Earth’s grav-ity using some a priori geophysical knowledge, then theresidual acceleration a r (cid:39) − z c (cid:2) a − (2 a + b ) a G + (cid:0) δ ,β (cid:1) ( a G ) (cid:3) . (21)Notice that a r is not continuous when β →
0, at which¯ w (cid:48) ( z ) becomes singular at z = 0. Suppose γ = 1, then a r (cid:39) − . × − m · s − ( a G /a ⊕ ) ( z /
10 m) in pureGR (Fig. 3b, red). In contrast, if the uniform gravityis entirely caused by the Higgs VEV (Fig. 3b, blue), then a r (cid:39) − . × − m · s − ( a M /a ⊕ ) ( z /
10 m). The neg-ative gradient means weaker gravity closer to ground,which is opposite to the Earth’s gradient due to its spher-ical shape. If one suspends a reference particle, another
FIG. 3. Exact [solid, Eqs.(17)-(19)] and approximate [dashed,Eqs. (20) and (21)] residual acceleration a r when initial posi-tion z = 0 (a) and velocity β = 0 (b), after an electric fieldis applied to cancel the acceleration at z = 0 and β = 0,assuming β = 0. In Newtonian gravity, a r is always zero.When GR [Eqs. (9)] and Higgs VEV [Eq. (14)] each con-tributes +50% of the uniform gravity (black), a r lies betweenthe extreme cases of pure GR (red, a G = a ⊕ ) and pure HiggsVEV gradient (blue, a M = a ⊕ ). Causes of gravity may there-fore be disambiguated by measuring the residual acceleration. test particle separated by z = 10 m will displace by∆ z ∼ − . t ∼ (cid:54)∝ m or a rulerwhose length is (cid:54)∝ m − . For example, using a plasmaoscillator, the clock frequency is ∝ m − / , provided thatthe plasma density, which introduces a separate dimen-sionful scale, is maintained. The Higgs VEV will thencause a blueshift on the GR redshift background.Finally, the depletion of the Higgs VEV by mattermay already be evidenced by distortions of galaxy spec-tralgraphs. For a given spectral line, a radial ( r ) massshifts compound with azimuthal ( θ ) Doppler shifts to give∆ f = f ( r )[ β s + β θ ( r ) sin ι cos θ ], where cβ s is the meanvelocity of the galaxy whose inclination is ι . Lopsidednessof this type have been observed while not satisfactorilyexplained for >
50% galaxies [55–64].In summary, nonconstant Higgs VEV can mimic somebut not all aspects of GR, and may explain, at least inpart, the observed dark energy, dark matter, Hubble ten-sion, and lopsidedness of galaxy spectrographs. Discrep-ancies between GR and Higgs VEV effects, albeit minus-cule on Earth, may be used to disambiguate causes ofgravity, which may involve both mechanisms.This work was performed under the auspices of theU.S. Department of Energy by Lawrence Livermore Na-tional Laboratory under Contract DE-AC52-07NA27344and was supported by the Lawrence Fellowship throughLLNL-LDRD Program under Project No. 19-ERD-038. ∗ [email protected][1] A. Einstein and M. Grossmann, Entwurf einer verall-gemeinerten Relativit¨atstheorie und einer Theorie derGravitation (Leipzig: Teubner, 1913).[2] A. Einstein, Die Feldgleichungen der Gravitation,Sitzungsber. K¨onigl. Preuss. Akad. Wiss. , 844 (1915).[3] C. W. Misner, K. S. Thorne, and J. A. Wheeler,
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