arXiv:1212.0263v1 [gr-qc] 27 Nov 2012 ∗ inesscalar sionless of fields. bodies. long- between such concept net transfer to a his energy induce apply range that fields not time-varying shown fields, does static is Unlike mass It falling invariant an problematic. is reference problem. fields this of solving introduction for essential The is mass for time. frames and coordinates mass. the relativistic rehabilitates of and concept GR, undervalued correspondence and relativity the special strengthens between harmonic approach and This isotropic of the metrics. choice the for the confirmed be of as independent to is coordinates, shown result new is special This from relativity. field mass relativistic Schwarzschild along for a expression falling old-fashioned in body a geodesic for proper a frame. relationship that reference position, mass/velocity mass way observer’s with The a local invariant a such is from perceived in mass as occur rest must measured) masses This (locally the with bodies. localized Therefore, of be [1]. would potential energy gravitational gravitational its with increases a lo- energy. the in over gravitational to perplexity of GR attempt long-standing calization an the with resolve is consistent to combination frames begin This are reference ideas observer field. pa- Schwarzschild Vera’s of this that accounting In be shows careful (GR). to a relativity thought per, general were widespread with they gained incompatible because not have possibly They acceptance, [1]. mass for concepts otasomtosbtenms eeec rms The frames. reference one- generalized mass a between then for transformations derived field, to is gravitational transformation static mass dimensional, A found. are lcrncades [email protected] address: Electronic h ae rceswt h olwn tp:Adimen- A steps: following the with proceeds paper The gravitational time-varying to concepts Vera’s Applying spatial with associated usually are frames Reference particle a of mass rest the that is ideas Vera’s of One fascinating introduced Vera Rafael ago years 30 Over ASnmes 42.q 42.v 03.30.+p field 04.20.Cv, static 04.20.-q, in numbers: exhibited PACS not is that bodies tha functi between demonstrates metric transfer scalar Robertson-Walker a the to as model localized this appr is proposed which The energy, coordinates. gravitational of relationship choice mass/velocity the show Observ of relativistic is independent It the mass. frames. exhibit rest reference field its mass different increases experience field would gravitational static a eateto hsc n srnm,Clfri tt Univ State California Astronomy, and Physics of Department nitrrtto fgnrlrltvt sdvlpdi whi in developed is relativity general of interpretation An β ab .INTRODUCTION I. sitoue;isipratidentities important its introduced; is oaie rvttoa nryi cwrshl field Schwarzschild a in energy gravitational Localized Dtd oebr2,2012) 27, November (Dated: Kentosh J ovninlteteto rvttoa nrya unlo- as energy gravitational of treatment conventional [9]. are GR energy of in enterprise.” forms be- inadmissible non-localizable the question’ that of this insists futility ‘answer Bondi Yet the to realized trying investigators to de- fore past were the effort wrong in and the voted to time answer enormous right Unhappily, energy- the question. gravitational for ‘local looking for who is formula momentum’ “Anybody magic words, a the for sought with looks Wheeler quandary and this resolve Thorne meaning- influ- to Misner, is an [8], In it Thus, textbook resides. energy ential 7]. gravitational [6, where much ask GR to After in less gravitational localized not particle. that is a arose energy of interpretation mass the gravitating deliberation, the en- potential to gravitational mass” ergy of “potential contribution term the describe the to proposed [5] Whittaker 1935, e fitrrtto n sitne ocmlmn GR. complement mat- to a intended is is work and This interpretation in fields. of seen static ter by not causes exhibited also not effects field effects gravitational causes time-varying a field fields, which static in electric equations, interpreta- time-varying Maxwell’s with proposed a As the imposed. on are limitations tion the and while studied fields field are Time-varying conserves Schwarzschild not. interpretation a does interpretation within conventional proposed mass the and motion. that energy oblique for shown and is relationship radial for It mass/velocity evaluated is The bodies falling metrics. other solution Schwarzschild and the to applied are transformations ti eie nexpression Ein- GR, an of derived development stein the During field? gravitational eaaetr nteeeg-oetmtensor energy-momentum a as the energy in attempts gravitational en- term Many defined gravitational [4] separate understand Tolman 3].) GR. better un- [2, in to an ergy see made in review coordinates been a have of (For choice pseudoten- way. the a seemly with is varies expression that That sor field. gravitational the lhuhtdyms eaiit emstse ihthe with satisfied seem relativists most today Although hthpest h nryue olf oyi a in body a lift to used energy the to happens What ∗ rmseilrltvt.Ti e eutis result new This relativity. special from nitisct eea eaiiy Applying relativity. general to intrinsic on s. r tdffrn rvttoa potentials gravitational different at ers ahpoie hsclepaainfor explanation physical a provides oach ievrigfilsidc e energy net a induce fields time-varying t htbde aln naSchwarzschild a in falling bodies that n hteeeg sdt itabd in body a lift to used energy the ch riy otrde aiona USA California, Northridge, ersity, I BACKGROUND II. t µν o h teseeg of stress-energy the for T µν In . 2 calizable, there remains considerable ambiguity in the interpretation of gravitational energy within GR. Many m authors believe that gravitational energy, and even mass, x are stored in fields [10–12]. Some conclude that gravita- g tional self-energy, which is a type of gravitational energy, contributes to the masses of bodies [13, 14]. Vera [1], Sav- ∆m Fall ickas [15], and Ghose and Kumar [16] interpret rest mass After Lift as depending on local gravitational potential, while Oha- m Impact v nian and Ruffini [10] argue that rest mass is a constant m m m m m m independent of space and time. Since energy cannot si- v multaneously be distributed in fields and contribute to ∆ particle masses, some of these views appear to be incom- m patible. Hayward states, “It comes as a surprise to many ( a ) ( b ) ( c ) ( d ) that there is no agreed definition of gravitational energy (or mass) in general relativity” [17]. A few dedicated researchers continue to study gravi- FIG. 1: (a) Two identical bodies with mass m at rest in a tational energy and related concepts such as quasi-local static gravitational field. (b) One is lifted and the lift energy increases its mass by ∆m to mx. (c) The lifted body falls back energy momentum (e.g., [2, 18–25]), driven partly by the to its starting point. While falling it maintains a constant belief, as stated by Mirshekari and Abbassi [26], that mass of mv = mx and has a greater mass than the body “one of the old and basic problems in general relativity at rest. (d) The fall is stopped and the body returns to its which is still unsolved is the localization of energy.” original mass m. Its extra mass ∆m is transferred to other bodies during the collision.

III. CONCEPT FOR GRAVITATIONAL ENERGY fields. However, it is shown later that the mass of a body cannot remain constant while falling in a time-varying As proposed or implied by others (e.g., [1, 15, 16]) it field, in which gravitational energy must be conveyed be- is hypothesized that the energy used to lift a body in tween bodies via gravitons and/or gravitational waves. a static gravitational field increases its rest mass. It is The essential element of Fig. 1 that remains applicable also assumed that the mass of a body remains constant in time-varying fields is that energy and mass are primar- while falling in a static field [1]. This interpretation is ily localized as part of the masses of bodies rather than illustrated in Fig. 1 with the following steps: The energy distributed in fields. 2 ∆E used to lift a body increases its rest mass by ∆E/c . For the purpose of this paper, no distinction needs to When the lifted body is allowed to fall from a higher be made between inertial mass, active gravitational mass, elevation, it maintains a constant mass during its free- and passive gravitational mass. They are assumed to be fall, equal to its mass at the point it began its fall. As the same [27, 28]. Also, the term “gravitational energy” the body passes its starting point with some velocity, it refers to the gravitational potential energy of a body and has a greater mass than an identical body at rest there, is not to be confused with the more general gravitational in general agreement with special relativity. When the energy-momentum tensor T µν . fall is stopped, the body loses its extra mass and returns As an example of the conservation of mass after an to the rest mass it had before being lifted. The mass that inelastic collision, thermal energy increases the velocity the body loses during its impact is transferred to other of the molecules of a body, with corresponding relativistic bodies (and to itself) in the form of vibrational energy, mass given by heat, elastic or inelastic energy, etc. Total mass would 2 always be conserved during lifting, falling, and collisions. mo mov /2 KE mv = ≈ mo + = mo + , (1) No mass or energy would be stored in the gravitational 1 − v2/c2 c2 c2 field. With this interpretation, the rest masses of all bodies so that thep mass of a falling body is in part converted would depend on position in a gravitational field. In ac- into the mass associated with additional kinetic energy cordance with the equivalence principle, any experiments KE of the molecules [29]. After the body cools off, it conducted at different elevations should be unable to de- returns to its original mass. tect any difference in mass, since all objects would be Although the terms “rest mass” and “relativistic mass” affected in exactly the same proportion. This interpre- have fallen out of favor [30–32], they are used herein out tation does for potential energy what special relativity of necessity. Here the term “rest mass” describes the did for kinetic energy – gives it a mass equivalent. The mass of a motionless body in any reference frame fixed history of various concepts of gravitational energy is a to a central mass in a static gravitational field. Since fascinating topic not covered herein. reference frames span different gravitational potentials, The concepts shown in Fig. 1 appear to work in static rest masses may vary with elevation. “Proper rest mass” 3 is the rest mass of a body when it is co-located with the from the perspective of observers who are motionless rel- observer measuring it; proper rest mass is presumed to ative to the field. Based on Einstein’s equivalence princi- be invariant. “Relativistic mass” refers to the mass of a ple, an observer at any location should measure the same body moving relative to an observer fixed with respect rest mass for a given body when it is at her location. This to the central mass, as perceived from that observer’s could be achieved experimentally by comparing the mass mass reference frame. Rather than dwell on semantic ar- of the body to the mass of a kilogram standard carried guments over the acceptability of the term “relativistic alongside. mass,” it is more productive to focus on physical argu- Consider an observer located at some arbitrary refer- ments and their consequences. ence elevation O. Let the proper rest mass of some body MTW [8] derived localized gravitational energy for the be mo. The mass of that body when it is atsome other el- Schwarzschild solution. However, their approach differs evation x, relative to the reference frame at O, is defined from Vera’s concepts [1]. to be mxo – an important notation for what follows. The Several variable mass theories of gravity have been difficulties of measuring mass at another location must be proposed [33–36]. However, they do not directly relate overlooked for now. When the body is at O, the observer changes in mass with potential energy. there will measure its mass to be moo ≡ mo. When the The standard model extension (SME) has been devel- body is carried to some other elevation x, an observer oped partly to study possible variations in fundamental there will measure its mass to be mxx = mo. constants [37–39]. In the SME, the relative masses of dif- Consider a static field causing a proper acceleration ferent materials might vary with gravitational potential, g(x) in the −x direction. Let x represent proper distance, in disagreement with GR. That possibility is not consid- which will generally differ from the coordinates used in ered herein. Modern versions of the E¨otv¨os experiments some solution to the field equations. Consider a body [40, 41] support the hypothesis that different materials with proper rest mass mo being slowly lifted against the are affected by gravitational fields in the same way. field. Then the proper energy required to lift the body some small distance dx is given by

IV. DIMENSIONLESS SCALAR β dE = mog(x)dx. (6)

In the local reference frame at x, the small increase in Because it is necessary to understand static fields be- mass as the body is lifted is given by fore exploring time-varying fields, this work begins with a detailed analysis of the Schwarzschild solution. For dE m g(x)dx dm = = o . (7) static gravitational fields it is useful to define the scalar xx c2 c2 dimensionless ratio βab as follows: From the reference frame at O, the body at x has mass dta β ≡ , (2) mxo, while an observer at x measures the mass of the ab dt b body to be mo. The ratio of those masses will apply to where dta is the time interval between two events mea- any body or mass increment at x. Therefore, the incre- sured at some point A and dtb is the time interval be- mental mass dm at x observed from O and from x are tween those two events measured from some other point related as follows: B, using clocks that are motionless relative to the field. dmxo mxo mxo If βab < 1, time passes more slowly at point A than at = = . (8) B. (In a static field it is easy to account for the time for dmxx mxx mo a signal to travel between A and B.) Combining (7) and (8) yields From the form of (2) the following identities for β can be found: dmxo g(x) = 2 dx. (9) mxo c βaa = βbb =1, (3) This can be integrated to yield the general mass trans- formation between elevations in a one-dimensional, static 1 βab = , (4) gravitational field: βba x and 1 mxo(x)= mo exp 2 g(x)dx . (10) c o βabβbc = βac. (5)  Z  This equation, also found in [15], indicates that the mass of a body lifted against the field to x is greater than V. GENERAL MASS TRANSFORMATION the mass of an identical body at O. Since there is no preferred elevation in a gravitational field, point O can The next step is to derive a general transformation for be selected to represent any point. The equation also rest mass in a static, one-dimensional gravitational field, shows that proper mass does not vary with elevation. 4

In a weak field, (10) reduces to mxo ≈ mo + (Φx − Besides using the identities, this can also be derived from 2 Φo)/c , where Φi is the classical gravitational potential the relative time rates at r and R. energy of the body. Since the mass transformation (10) is based on proper Next the general transformations between mass refer- acceleration g(x), the next step is to find proper acceler- ence frames can be found. Consider the general case of ation relative to a motionless observer in a Schwarzschild two elevations A and B. Let O → A and x → B. From field. For radial motion, Hobson et al. [42] find the fol- the form of the preceding equation, it can be shown that lowing geodesic equation of motion: any rest mass mo at B relative to the mass reference frame at A, and vice-versa, satisfies identities that are GM r¨ = − , (18) similar to those for β, so that r2 m m aa = bb =1, (11) wherer ˙ ≡ dr/dτ. In the general case, τ is the time mea- mo mo sured by a clock that may be either moving or stationary. − In the case of a stationary clock at r, the terminology m m 1 → ab = ba , (12) τ tr is used to differentiate between those two cases. m m For a motionless clock at r, dt/dτ ≡ dt∞/dt =1/β ∞. o  o  r r and Coordinate r in the Schwarzschild metric has little m m m physical significance. Of greater relevance is the proper ab bc = ac , (13) distance given by ds, which in the general case may cor- mo mo mo respond to a moving meter stick or one that is motionless where mab, for example, is the mass of a body at A, with relative to the central mass. To distinguish between those proper rest mass mo, relative to the mass reference frame two possibilities, variable x is used to denote proper dis- at B. Observers at different gravitational potentials ex- tance in the reference frame of a motionless observer at perience different mass reference frames that are related r. By setting dt = dθ = dφ = 0, letting ds → dx in (14), by these formulas. and using the appropriate sign for a spacelike spacetime interval, the following relationship is found:

VI. MASS TRANSFORMATION OF THE dr = βr∞dx, (19) SCHWARZSCHILD SOLUTION which relates proper distance dx to the radial coordinate Next, the mass transformation is applied to the exte- distance dr. Using the resulting formula for dx in place rior Schwarzschild solution of GR, described by the fol- of dr in (18), lowing metric: 2 2 d dx dβr∞ dx d x dr ∞ ∞ 2 2 2 2 − − 2 2 − 2 2 2 r¨ = βr = + βr 2 . (20) ds = c βr∞dt 2 r dθ r sin θdφ , (14) dτ dτ dτ dτ dτ βr∞   where Consider a body at the instant it is released to free- fall, before it has attained appreciable velocity. Then 2 βr∞ = 1 − 2GM/c r. (15) dτ ≈ dtr, the proper velocity is dx/dτ ≈ 0, and only the The coordinate time dtpis the time interval of a station- right hand term in (20) contributes. Using the resulting ary clock at infinity and ds = c dτ, where dτ is proper equation in combination with (18) and (19), the proper time of a clock corresponding to the spacetime interval acceleration of gravity for a slowly falling body at r in a ds. Setting dr = dθ = dφ = 0 in (14) for a motionless Schwarzschild field is given by clock at r yields βr∞ = dtr/dt∞, where dτ → dtr is the 2 time interval of a stationary clock at r and dt∞ = dt is ≡−d x ≈ GM g(r) 2 2 . (21) the time of a stationary clock at infinity. Thus, βr∞ is dtr βr∞r the ratio of the time rates at r and infinity, correspond- ing to the definition for β in (2). That is the origin of This is the acceleration that must be resisted when slowly the subscripts in (15). lifting a body in the field. Note that g(r) is positive Equation (15) provides β at r relative to an observer towards the central mass. at infinity. At some other radius R, Suppose a small body with proper rest mass mo is lifted radially from r to R in a Schwarzschild field. The − 2 βR∞ = 1 2GM/c R. (16) mass of the body at R, relative to the reference frame at From identities (4) andp (5) an expression for the β’s r, can be found by substituting (19) and (21) into (10), relating elevations r and R can be found: which yields

2 R βR∞ 1 − 2GM/c R GM dr βRr = = . (17) mRr = mo exp 2 2 2 . (22) β ∞ 1 − 2GM/c2r c β ∞r r s Zr r ! 5

The surprisingly simple solution to this integral is mass of a body that has been lifted from r to R. Based on the concept in Fig. 1, if that body is allowed to fall βR∞ from R back to r, it will maintain the same mass during mRr = mo = βRrmo. (23) βr∞ its fall. It will therefore arrive at r with the relativistic mass mv = mRr given by (23). Since βRr > 1 if R > r, the mass of the lifted body at In GR it is straightforward to derive the velocity of a R is perceived to be greater than mo from the reference body falling radially from R to r in a Schwarzschild field. frame at r. In a weak field, mRr is its rest mass plus The radial geodesic equations of motion [42] are (or minus) the mass equivalent of the change in classical potential energy. r˙2 = c2(k2 − 1)+2GM/r, (27) If r is chosen to be the reference frame at an infinite distance from the central mass and if R → r, then the and relative mass of a body at any point r perceived from the 2GM 1 − t˙ = k. (28) reference frame at infinity is given by c2r   mr∞ = βr∞mo. (24) Expressingr ˙ in terms of dr/dt and setting constant k so that the coordinate velocity dr/dt = 0 at R, the Thus, stationary bodies in the vicinity of a central mass coordinate velocity of the body when it arrives at r is have a decreased mass relative to a reference frame at infinity. It is apparent that βr∞ plays the role of scale 2 dr 2 βr∞ = cβ ∞ 1 − . (29) factor for mass as well as time. dt r β2 The preceding results for the Schwarzschild solution s R∞ suggest the following general transformation for rest mass Using prior formulas, the proper velocity vr of a body in any static gravitational field: at r can be related to its coordinate velocity as follows:

mxo = βxomo. (25) ≡ dx 1 dr vr = 2 , (30) dtr βr∞ dt This generalization, offered without proof, does not de- pend on the choice of coordinates; it is not used again where vr is measured with respect to proper coordinates in this paper. The proof of (25) for any static field g(x) x fixed relative to the central mass. The preceding two requires topics not covered here. equations can be combined to yield the proper velocity Most studies of localization focus on the gravitating of a body falling from R, measured at r by a motionless mass rather than test particles. Nevertheless, mass ref- observer there: erence frames reveal an interesting property of the grav- 2 itating mass in a Schwarzschild field. In some ways, the − βr∞ − −2 vr = c 1 2 = c 1 βRr . (31) Schwarzschild metric is written from the perspective of s βR∞ q a reference frame at infinity: Coordinate time t repre- sents a clock at infinity and radial distance r converges Using (23) to provide the locally measured mass of to flat space coordinates at infinity. Therefore, the cen- the falling body mv = mRr, it is found that the relation- ship between proper velocity and total mass of the falling tral mass at the origin should be M ≡ Mo∞, its mass from the reference frame at infinity. From a reference body at r is given by frame at r, all masses appear increased from (23), so mo mv = . (32) that Mor = Mo∞/βr∞. Substituting this into (21) yields − 2 2 1 vr /c

GMor This is the well known relativisticp mass/velocity relation- g(r)= , (26) r2 ship of special relativity, derived for a body falling in a Schwarzschild field. where r is the proper circumference divided by 2π. Thus, Newton’s law of gravity is preserved for any local observer if the magnitude of the central mass in his reference frame VIII. OTHER INITIAL CONDITIONS is used. This result is independent of the choice of coor- dinates. The mass/velocity relationship of (32) was derived for a body released from rest at R. What if some initial, VII. RELATIVISTIC MASS OF A FALLING proper radial velocity vR is imparted to the body at R BODY as it begins its fall? Then an observer at R will perceive the body to have an increased mass as it begins its fall, as given by Now that the mass transformation has been derived for a Schwarzschild field, the mass/velocity relationship for mo mvR = . (33) − 2 2 a falling body can be found. Equation (23) provides the 1 vR/c p 6

From the reference frame at r, the mass of the body This discussion applies only to static fields. The time- at R will be increased above its rest mass mRr in the varying field created by the expansion of the universe same proportion. That is the value of the body’s mass causes a cosmological redshift that is distinct from that that remains constant during the fall from R to r, and of static fields. will be measured as mv when it arrives at r. Using the equation of motion (27) and other equations previously derived, it can be shown that, regardless of any initial X. OTHER METRICS FOR A CENTRAL MASS velocity imparted to the body at R, the mass/velocity relationship of (32) applies when the body arrives at r. Consistent with Birkhoff’s law [50], many metrics de- scribe the field of a central mass, each related to the Schwarzschild solution by a coordinate transformation. IX. GRAVITATIONAL REDSHIFT Two such metrics are the isotropic and harmonic met- rics [51]. The isotropic metric for a central mass can be For static fields there are two distinct physical expla- written in the form nations for gravitational redshift that can be found in the 2 2 2 2 2 2 2 2 2 2 2 ds = c β ∞dt − α (dr + r dθ + r sin θdφ ), (35) literature. The most common view is that the redshift r I I I is caused by a loss (gain) in the energy of a photon as it where climbs (descends) in a gravitational field. Another view, 2 argued by Okun, Selivanov and Telegdi [43, 44] and noted βr∞ = (1 − GM/2c rI )/α, (36) by others (e.g., [45, 46]) is that the redshift is caused by a change in the energy levels of lifted atoms. These view- and points are incompatible; if both were true the redshift 2 would be doubled. α =1+ GM/2c rI . (37) The concept of mass reference frames supports the If a body is lifted from rI to R in the isotropic metric, view that gravitational redshift is caused by a change and then allowed to fall back to rI , the mass/velocity in the energy levels (masses) of lifted atoms. The en- relationship of (32) is found at rI . ergy of a photon would not change while traveling along If this exercise is repeated for the harmonic metric [51], a null geodesic in a static gravitational field [1, 15]. If that result is found once again. What this confirms is a photon is emitted at elevation O with proper energy that mass reference frames are independent of the choice Eo, it will arrive at a higher elevation x with the same of coordinates, as should be expected. It can also be energy. However, all stationary masses and energies at shown that at the same point in space, as determined by x are increased relative to O. To an observer at x, the well-known coordinate transformations between the three photon arrives with a reduced energy compared to a pho- metrics (e.g., [51]), both βr∞ and the local gravitational ton at x emitted by the same process. Using a uniform ≈ acceleration g(r) are identical for all three metrics. field approximation of g(x) go and E = hf, it can be What this suggests is that, in static fields, gravita- shown that a photon with proper energy Eo emitted at tional energy is stored as part of the masses of bodies O arrives at x with a frequency shift of in accordance with the simple scalar function βab, which transforms in a well-behaved way and is independent of ∆f g x = − o , (34) the choice of coordinates. f c2 relative to the frequency the photon would have if it were XI. NON-RADIAL MOTION emitted at x with proper energy Eo. The proper fre- quency of the photon at x is established by its proper energy there, in combination with Planck’s constant, and Up to this point, only radial motion relative to a cen- is not an inherent property of the photon. Experimen- tral mass has been considered. Next it is shown that mo- tal evidence of the local position invariance of Planck’s tion in any oblique direction is not only consistent with constant [47] is consistent with this interpretation. the concept of mass reference frames, but corresponds to the Schwarzschild metric. A review of gravitational redshift experiments (e.g., Consider a stationary observer fixed at some radius r [48]) indicates that they are unable to distinguish from a central mass. To that observer, special relativity whether photons lose energy during travel, or appear to and (32) indicate that a moving or orbiting body at r be redshifted due to changes in mass/energy reference should have a mass m of frames. The proposed interpretation also satisfies the vrr thought-experiment of Nordtvedt on the conservation of mo mvrr = , (38) energy in gravitational redshift experiments [49]. The − 2 2 1 vr /c proposed existence of mass reference frames appears to be consistent with predictions and observations of gravi- where mvrr denotes relativisticp mass mv of the body at tational redshift. r in the reference frame at r. From a reference frame 7 at infinity, the mass of the moving body at r appears obtained when the corresponding expression for dx2 is smaller, since all masses at r appear decreased relative used. Although this is not an independent derivation of to those at infinity by a factor βr∞, from (24). From that the metrics, it demonstrates the correspondence between frame, the mass of the moving body at r is perceived to mass reference frames and the metrics of GR. Since this be result is based on (38) it also indicates that a body mov- ing in any oblique direction will exhibit the mass/velocity βmo mvr∞ = , (39) relationship of special relativity in a local frame. − 2 2 1 vr /c where the subscripts on βrp∞ are dropped for now. If the XII. FINAL SUPPORTING ARGUMENT FOR body is in free-fall around the central mass, moving in STATIC FIELDS any direction with no other external forces, its total mass will remain constant during its fall or orbit. Therefore, The derivation of (32) confirms that mass and energy the body’s total mass from the reference frame at infinity are conserved during a lift and fall cycle under the pro- is constant as follows: posed interpretation. That provides a compelling argu-

mvr∞ β ment to support the existence of mass reference frames: = K = , (40) As shown below, the prevailing interpretation of gravita- mo 1 − v2/c2 r tional energy in GR does not conserve energy and mass where K is a constant of the motion.p This formula makes during a lift and fall. it easy to find the velocity at any radius r. If K > 1 Consider a body being slowly lifted in a Schwarzschild the body has sufficient relativistic mass – and associated field. The mass equivalent of the proper lift energy ex- velocity – to escape the central mass. If K < 1 the orbit pended over a small distance dx is given by (7). If to- will be bound. This result is similar to that of [52]. tal mass is indeed invariant between all elevations, then mxo = mo, contrary to (9). Then the mass equivalent Using the expressions vr ≡ dx/dtr and dtr = βdt, (40) can be rearranged to yield of the energy used to lift a body from O to x would be given by 2 4 2 c β dt 2 2 2 2 x = c β dt − dx , (41) mog(x)dx K2 ∆m = . (46) c2 Zo where dt is the time of a clock at infinity. Let dτ represent the time of a clock on the moving body. Those two times If this is integrated in a Schwarzschild field, the total are related by mass equivalent of the energy expended while lifting the body from r to R would be given by dτ = β 1 − v2/c2. (42) ∆m = m ln β . (47) dt r o Rr p Using (40) and (42), it can be shown that the left hand In weak fields, mo + ∆m approaches mRr in (23). side of (41) is c2dτ 2 = ds2, so that When the body falls from R to r it must arrive with a relativistic mass given by (32) in the reference frame of a ds2 = c2β2dt2 − dx2. (43) stationary observer at r. It is easy to show that the lift energy/mass and the additional relativistic energy/mass As previously defined, dx is proper length in the mo- acquired by the body when it returns to r are not con- tionless reference frame at r. For any given metric, dx served, as follows: can be related to the coordinates used in that solution. mo − For example, for the standard form of the Schwarzschild ∆m< mo. (48) 1 − v2/c2 solution r p dr2 It takes less energy/mass to lift a body than arrives back dx2 = + r2dθ2 + r2 sin2 θdφ2. (44) at the starting point. Thus, without the use of mass β2 reference frames, energy and its mass equivalent are not This can be found from the metric at a synchronous in- conserved within a lift/fall cycle. This does not indicate stant in time (dt = 0), which gives the spacelike space- any flaw in GR, but suggests that mass reference frames time interval ds2 → dx2 when the appropriate sign is are necessary to complement the theory. used. Substituting this into (43) yields XIII. MACH’S PRINCIPLE dr2 ds2 = c2β2dt2 − − r2dθ2 − r2 sin2 θdφ2. (45) β2 To the extent that distant stars contribute to the lo- This is the Schwarzschild metric of (14). Other metrics, cal gravitational potential at any point, the proposed in- like the isotropic and harmonic metrics, can similarly be terpretation implies that the magnitude of the inertia of 8 any body is directly influenced by masses throughout the 2. Conversely, the rest mass of M2 would necessarily in- universe. Thus, the proposed interpretation brings GR crease. That interpretation would require the rest masses into closer harmony with Mach’s principle. Nevertheless, of all bodies and elementary particles to vary with their the concepts developed herein are more likely to fuel the prior motion in time-varying gravitational fields. In the recurring debate on Mach’s principle than to resolve it. absence of experimental evidence of an appreciable vari- ability in the rest masses of protons or electrons, it is evident that Vera’s interpretation of a falling body does XIV. TIME-VARYING FIELDS not work in a time-varying field. Fortunately, all is not lost. The most essential of Vera’s This paper focuses on static fields, which encompass concepts is that gravitational energy is stored as part the most important tests of GR. Nevertheless, it is im- of the masses of bodies rather than in empty fields. In portant to show how the proposed interpretation is lim- Fig. 2, the energy expended by the winch travels directly ited in time-varying fields. Such fields introduce issues to M2 via the cable. Half of that energy then travels that are discussed only briefly here. through space almost immediately to M1, limited only The analysis of time-varying fields begins with the sim- by c. No energy would be stored in fields over the long- ple thought-experiment shown in Fig. 2. Consider two term. masses M1 = M2 located deep in intergalactic space and What can be deduced from this exercise is that gravita- separated by distance d. Initially, both masses are ap- tional energy must travel through space in a time-varying proximately co-moving with the cosmological fluid. The field. Bondi described the movement of energy across the attractive gravitational force between the bodies is F . empty space between two bodies [9]. In static fields, en- Imagine that a long cable is attached to M2, with the ergy may be exchanged between gravitating masses, but other end attached to a winch W secured to a distant, the net change in the mass of a test particle during free- massive object. The winch pulls on the cable with a ten- fall should be zero. Thus the proposed interpretation sion of T = 2F , so that mass M2 moves away from M1 of gravitational energy is compatible with time-varying at the same rate that M1 falls towards M2. Both masses fields, but is limited accordingly. This approach is consis- will accelerate towards W while maintaining distance d tent with the transport of energy by gravitational waves, between them. Once they have achieved some large ve- and also suggests that gravitons, if they exist, should locity v, let mass M2 be pulled a great distance away from convey mass/energy. The transfer of energy in Fig. 2 is M1, leaving M1 moving at velocity v in empty space. fundamental and does not arise from the concept of mass reference frames. To show how the proposed interpretation works in time-varying fields, it is next applied to the most impor- d tant cosmological metric of GR – the Robertson-Walker metric, which can be written in the following form: F F T = 2F M M W 1 2 ds2 = c2dt2 − a2(t)[dχ2 + sin2 χ(dθ2 + sin2 θdφ2)], (49) v v where t is the cosmological time of all observers co- moving with the cosmological fluid, a(t) is the scale factor representing the expansion of the Universe, and dχ rep- FIG. 2: Two masses M1 = M2 in empty space, with a mutual resents radial coordinate distance. Proper distance dx in gravitational force F . A long cable pulls on M2 with a tension the radial direction is related to dχ by dx = a(t)dχ at T ≈ F 2 , causing both masses to accelerate to the right at the any synchronous time t. same rate, maintaining their separation d. After some time, Suppose that at some time t a particle is given an M2 is pulled away from M1, leaving M1 with final velocity 1 v. Energy has traveled from the winch W to M2, and then initial proper radial velocity V relative to the cosmologi- through empty space to M1. cal fluid,. At some later time t, it will arrive at a second point with proper velocity vr ≡ dx/dt. From the geodesic equations of motion (e.g., [42]), the proper radial velocity In this example, body M1 has been accelerated by of the body at t can be found: gravity to velocity v, imparting relativistic energy/mass dx V to the body. Where has that energy come from? If Vera’s vr(t)= = 2 . (50) dt V 2 a (t) V 2 interpretation of falling bodies from Fig. 1 is applied, 2 2 − 2 c + a (t1) 1 c then the total mass remains constant and the original q
rest mass mo has been converted into relativistic mass As expected, the equation confirms that if a particle mv = mo. If the body’s motion is then stopped, it will starts out with an initial velocity V = 0 relative to the lose the relativistic part of its mass and drop to a new cosmological fluid it will continue to move with the cos- ′ ′ rest mass of mo

As previously discussed for the Schwarzschild solution, Equation (51) provides another way to calculate suppose a particle is slowly “lifted” between two points. cosmological redshift. If mV is the relativistic mass What (50) indicates is that no matter how small the ini- of a particle at time t1 when it has initial velocity V , tial velocity V > 0, the particle’s proper velocity will then the limit of mv(t)/mV as V → c is a(t1)/a(t), never subsequently fall to zero. It will forever continue which in terms of energy corresponds to the expected to drift slowly across the Universe. This means that all cosmological redshift of a photon [42]. points in the expanding universe have the same grav- itational potential. There is no gravitational force or XV. CONCLUSIONS pseudo-force to lift against when slowly moving a parti- cle. Therefore, all observers co-moving with the cosmo- The proposed interpretation of gravitational energy logical fluid experience the same mass reference frame. provides a means to explore the disposition of gravita- However, (50) shows that as the universe expands, the tional energy in solutions to the field equations. The proper velocity of of the particle decreases monotonically localization of gravitational energy in static fields is ex- below its initial proper velocity V . Therefore, its rela- plained by simple concepts in which the energy is local- tivistic mass will decrease with time, relative to cosmo- ized with the masses of bodies. Observers at different logical observers. Using (50) the relativistic mass of a gravitational potentials would experience different mass particle as a function of time is reference frames. With this approach, it is shown that the relativistic mass of a falling body can be calculated 2 2 a (t1) V directly from GR. Thus, the correspondence between gen- mv(t)= mo 1+ , (51) s a2(t) (c2 − V 2) eral and special relativity is enhanced. Unlike the pseu- dotensor tµν , this interpretation is independent of the where mo is its rest mass. Thus the relativistic mass choice of coordinates. The use of mass reference frames decreases with time. The change in relativistic mass with also conserves mass and energy during a lift/fall cycle, time can be found from (50) and (51) unlike the concept of invariant mass. Inasmuch as there currently exists no widely-accepted treatment of localized 2 dmv vr a˙(t) gravitational energy in GR, it is hoped that the proposed = −mv . (52) dt c2 a(t) interpretation will be considered worthy of further study and discussion. Much work remains to be done. This loss of relativistic mass is directly proportional to Many attempts have been made to incorporate a scalar the rate of expansion of the Universe. In a static universe, field into gravitational theory. Examples are the scalar- a˙ (t) = 0 and there is no loss of mass. This result is tensor theory of Brans and Dicke [53, 54] and the dilaton consistent with Vera’s interpretation that the mass of field [55]. What this paper suggests is that, in static a falling body remains constant in a static field, while fields, GR already includes an intrinsic scalar field β requiring the transfer of mass in a time-varying field. It is ab that accounts for the energy of the gravitational field. important to recognize that the loss of mass in (52) does not arise from the concept of mass reference frames, but Unlike static fields, time-varying fields require a net is a direct consequence of the Robertson-Walker metric exchange of energy and mass between gravitating bod- and is implicit in GR. ies. With this interpretation, gravitational waves and/or So what happens to the relativistic mass lost by a body gravitons would convey energy and mass. moving relative to the cosmological fluid? The concepts In Newtonian gravity, potential energy allows total en- developed herein suggest that mass/energy must travel ergy to be conserved when lifting a body. The equiva- through space to other bodies, most likely at the speed lence of a small amount of mass to a large amount of en- of light. For the Robertson-Walker metric, the receiving ergy allows total energy and mass to be conserved with- bodies would be the masses comprising the expanding out it, consistent with everyday experience. Universe. Similarly, this approach can be applied to any metric to quantify the mass/energy lost or gained by falling bodies. The concepts developed herein only begin to explore this References topic.

[1] Vera R A 1981 Int. J. Theor. Phys. 20 19 [5] Whittaker E 1935 Proc. R. Soc. London A 149 384 [2] Szabados L B 2009 Liv. Rev. Rel. 12 4 [6] Landau L and Lifshitz E 1959 The Classical Theory of (www.livingreviews.org/lrr-2009-4) Fields (Reading: Addison-Wesley) [3] Baryshev Y 2008 Practical Cosmology 1 276 [7] Dirac P A M 1975 General Theory of Relativity (New [4] Tolman R 1930 Phys. Rev. 35 875 York: John Wiley and Sons) 10

[8] Misner C, Thorne K and Wheeler J 1973 Gravitation [33] Dicke R H 1962 Phys. Rev. 125 2163 (San Francisco: Freeman) [34] Malin S 1974 Phys. Rev. D 9 3228 [9] Bondi H 1990 Proc. R. Soc. Lond. A 427 249 [35] Hoyle F 1975 Astrophys. J. 196 661 [10] Ohanian H and Ruffini R 1994 Gravitatation and Space- [36] Bekenstein J D 1977 Phys. Rev. D 15 1458 time (New York: W. W. Norton) [37] Colladay D and Kosteleck´yV A 1997 Phys. Rev. D 55 [11] Stephani H 1990 General Relativity (Cambridge: Cam- 6760 bridge University Press) [38] Colladay D and Kosteleck´yV A 1998 Phys. Rev. D 58 [12] Pauli W 1958 Theory of Relativity (New York: Pergamon 116002 Press) [39] Kosteleck´yV A 2004 Phys. Rev. D 69 105009 [13] Baeβler S, Heckel B R, Adelberger E G, Gundlach J H, [40] Dent T 2008 Phys. Rev. Lett. 101 041102 Schmidt U and Swanson H E 1999 Phys. Rev. Lett. 83 [41] Wagner T A, Schlamminger S, Gundlach J H and Adel- 3585 berger E G 2012 Class. Quantum Grav. 29 184002 [14] Uzan J-P 2003 Rev. Mod. Phys. 75 403 [42] Hobson M P, Efstathiou G P and Lasenby A N 2007 [15] Savickas D 1994 Int. J. Mod. Phys. A 9 3555 General Relativity, an Introduction for Physicists (Cam- [16] Ghose J K and Kumar P 1976 Phys. Rev. D 13, 2736 bridge: Cambridge University Press) [17] Hayward S A 1994 Phys. Rev. D 49 831 [43] Okun L B 2000 Mod. Phys. Lett. A 15 1941 [18] Chang C-C, Nester J M and Chen C-M 1999 Phys. Rev. [44] Okun L B, Selivanov K G and Telegdi V L 2000 Am. J. Lett. 83 1897 Phys. 68 115 [19] Wu M-F, Chen C-M, Liu J-L and Nester J M 2011 Phys. [45] Feynman R 1995 Feynman Lectures on Gravitation ed Rev. D 84 084047 Hatfield B (New York: Addison-Wesley) [20] Wang M-T 2011 Class. Quantum Grav. 28 114011 [46] Hayward R 1967 Handbook of Physics ed Condon E and [21] Mitra A 2010 Gen. Rel. Grav. 42 443 Odishaw H (New York: McGraw-Hill) [22] Zhang X 2009 Class. Quantum Grav. 26 245018 [47] Kentosh J and Mohageg M 2012 Phys. Rev. Lett. 108 [23] Liu C-C M and Yau S-T 2003 Phys. Rev. Lett. 90 231102 110801 [24] Frauendiener J and Szabados L B 2011 Class. Quantum [48] Pound R V and Snider J L 1964 Phys. Rev. Lett. 13 539 Grav. 28 235009 [49] Nordtvedt K 1975 Phys. Rev. D 11 245 [25] Katz J 2005 Class. Quantum Grav. 22 5169 [50] Birkhoff G D 1923 Relativity and Modern Physics (Cam- [26] Mirshekari S and Abbassi A M 2009 Mod. Phys. Lett. A bridge: Harvard University Press) 24 747, arXiv:0808.0179 [51] Bodenner J and Will C M 2003 Am. J. Phys. 71 770 [27] Rosen N and Cooperstock F I 1992 Class. Quantum Grav. [52] Augousti A T, Radosz A and Ostasiewics K 2011 Eur. J. 9 2657 Phys. 32 535 [28] Roche J 2005 Eur. J. Phys. 26 225 [53] Brans C and Dicke R H 1961 Phys. Rev. 124 925 [29] Carlip S 1998 Am. J. Phys. 66 409 [54] Barraco D and Hamity V 1994 Class. Quantum Grav. 11 [30] Okun L B 1998 Eur. J. Phys. 19 403 2113 [31] Adler C G 1987 Am. J. Phys. 55 739 [55] Fujii Y 2003 Prog. Theor. Phys. 110 433 [32] Okun L B 2009 Am. J. Phys. 77 430