The concept underlying conservation of energy in a force field is that the work done by particles moving against the force represents energy removed from particles and added to energy in the field. For electromagnetic fields and charged particles, the transfer of energy between charges and local fields is described by Poynting's theorem. However, a similar local model for gravitational energy is not usually taught. In this paper, I derive the equations describing local conservation of energy in Newtonian gravity, discuss the well-known non-uniqueness of various terms, and show how comparison with Poynting's theorem provides insight into conditions for physical significance.

Understanding and quantitatively describing conservation of energy, an important part of learning physics, has been the subject of renewed interest since the publication of Framework for Science Education1 and the associated development of Next Generation Science Standards (NGSS).2 These reports focused attention on a long-standing issue: Is energy associated with force fields (e.g., electromagnetic, gravitational) better represented in terms of potentials or of the fields themselves?3,4 Here, I discuss energy in purely classical (non-relativistic) gravitational fields and show that it involves subtle aspects not always adequately treated in textbooks.

The work-energy theorem as usually stated applies to complete isolated systems and must be generalized if we deal with extended continuous media, including not only solids or fluids but also gaseous systems described by local quantities that represent averages over small volumes containing many particles. As illustrated in the examples described later in this paper, the work that is done locally by the flow against gravity cannot, in general, be equated to the local change of energy in the field because energy can be transported to other locations. To describe, this quantitatively requires a local energy equation. This process is well understood for electromagnetic energy, where Poynting's theorem is the local energy equation, but an equation that would play the same role (although not necessarily having the same mathematical form) in gravity as Poynting's theorem does in electromagnetism is hardly ever discussed explicitly.

The whole issue of how to describe changes of gravitational energy within the classical Newtonian context seems to have received little attention. Textbooks sidestep the problem by discussing Newtonian gravitational energy only in the integral form for isolated systems,5 or they do not include gravity in the discussion of energy conservation either because a relativistically correct treatment is aimed at but the gravitational terms have been formulated only in a nonrelativistic approximation,6 or, on the contrary, because gravity is to be treated at the general relativistic level and is purposely left out of the preliminary Newtonian description,7 or they simply do not mention gravity at all in the discussion of energy.8 My purpose here is to try to fill this gap in textbook coverage.

As final revisions to this paper were being made, I learned about another paper to be published in this journal that examines a general description of gravitational fields, including energy flows, from a relativistic “gravitoelectromagnetic” perspective where the field equations are formulated in analogy to Maxwell's equations.9 In contrast, the approach taken in this paper is a classical Newtonian model where fields respond instantly to the motion of charges and masses. Readers are encouraged to examine both papers for a full picture of this issue.

As a simple example of the issues that arise when changes in gravitational potential energy occur, consider water in a pipe pushed up against gravity. This arrangement was discussed as a thought experiment long ago by Bridgman,10,11 and it is of current practical importance in the pumped-storage system (Fig. 1) of some hydroelectric power plants.12 Two water reservoirs are located at different heights with water being pumped from the lower to the upper. The work is done by the pump; where does the energy that is being expended to pump the water go? A simple answer is that the energy expended in lifting a mass m against Earth's gravity g0 to a mean height h above ground is stored in the mass m as the change of its gravitational potential energy.13 However, this explanation is unsatisfactory if we think of the Earth as part of the system. As shown in the  Appendix, the change of gravitational potential energy is in fact divided equally between the water and the Earth, even though the work done on the Earth is negligible.14

Fig. 1.

Sketch of pumped-storage system.

Fig. 1.

Sketch of pumped-storage system.

Close modal

The work done per unit volume by the mass flow against gravity is given by

$work per unit volume =−ρV·g,$
(1)

where $ρ=$ mass density, $g=$ gravitational field, and $V=$ bulk flow velocity, all functions of position r and time t. If energy is conserved, the total work done must equal the rate of change of gravitational energy within the entire system,

$−∫d3r ρV·g=dUGdt.$
(2)

Since the total gravitational energy UG is a quantitative property of the system, expressible as the volume integral of gravitational energy per unit volume (energy density) uG,

$UG=∫d3r uG.$
(3)

Equation (2) can be written as

$−∫d3r ρV·g=∫d3r ∂uG∂t.$
(4)

With the integrals on both sides extending over the entire system (which in some cases may be over all space), this is a global equation. It becomes a local equation not when the two integrands are equal but rather when they differ by at most the divergence of a vector $SG$,

$−ρV·g=∂uG∂t+∇· SG,$
(5)

subject to the condition that the surface integral of the component of $SG$ normal to the outer boundary of the system vanishes,

$∫dA n̂· SG=0.$
(6)

The quantity $SG$ is a vector representing energy transport from one location to another; it is the local gravitational energy flux density, that is, the amount of energy transported across a directed unit area. Equation (5) expresses local conservation: Energy supplied locally by the work done per unit volume either changes the field energy per unit volume or flows out into adjacent volumes; the form of the equation (RH side consisting of a time derivative plus a divergence) ensures that the sum of the two equals the work done. Global conservation of energy expressed by Eq. (4) requires the boundary condition Eq. (6) of no net flow of energy out of or into the system. The energy flux density SG is not mentioned in most standard texts.

An energy equation in the form of Eq. (5) expressing the work done as the time derivative of an energy density plus divergence of an energy flux density is a necessary condition for conservation of energy in the interaction between force fields and matter, not just for Newtonian gravity but also, with the appropriate redefinition of work done, for force fields quite generally. For electromagnetism, the work done is given by electric field times current density, and the energy equation is the familiar Poynting's theorem,

$−E·J=∂∂t(E2+B28π)+∇·(c4π E×B) Gaussian units,$
(7)
$−E·J=∂∂t(ϵ0E22+B22μ0)+∇·(E×Bμ0) SI units,$
(8)

where $J=$ electric current density, $E=$ electric field, $B=$ magnetic field, and $c=$ speed of light. The reason for showing the equation in both Gaussian and SI units will become apparent in Sec. VI. That such an expression for $E·J$ can be derived from Maxwell's equations shows that energy is conserved in electromagnetic interactions and also provides definitions for field energy density and energy flux density. Deriving the analogous equation (5) for $−ρV·g$ from the gravitational field equations is needed to show that the energy is conserved in Newtonian gravity and to provide definitions for uG and $SG$.

Expressions for uG and $SG$ are derived here in Sec. III. They are not unique: Only the rate of work per unit volume is uniquely defined in terms of measurable physical parameters, but there are several matched pairs uG and $SG$, with any of which Eq. (5) describes energy conservation quantitatively; Poynting's theorem is similarly non-unique.15,16 The significance of various expressions is discussed in Sec. IV. The Newtonian energy equation is illustrated by examples in Sec. V and is compared with Poynting's theorem in Sec. VI.

The derivation of the local energy equation of Newtonian gravity involves the field equations,

$∇·g=−∇2Φ=−4πGρ,$
(9)
$∇×g=0 g=−∇Φ$
(10)

($Φ=$ gravitational potential) and the flow continuity (mass conservation) equation,

$∇·ρV+∂ρ∂t=0.$
(11)

Equations (9) and (11) imply that

$∇·(ρV−14πG ∂g∂t)=0.$
(12)

Expressing work per unit volume as

$−ρV·g=ρV·∇Φ=∇·(ρVΦ)−Φ∇·ρV,$
(13)

and invoking Eq. (11) yields the equation

$−ρV·g=Φ∂ρ∂t+∇·(ρVΦ),$
(14)

which does not have the conservation form of Eq. (5), because the first term on the right side is not a simple time derivative of a single quantity. Two ways of modifying this term lead to two different forms of the energy equation. The following derivations will make extensive use of Green's identities:

$∇·(a∇b)=∇a·∇b+a∇2b$
(15)

and

$∇·(a∇b−b∇a)=a∇2b−b∇2a.$
(16)

Use Eq. (9) to replace $∂ρ/∂t$ by $∇·∂g/∂t$, obtaining

$Φ∂ρ∂t=−Φ∇·(14πG ∂g∂t)=−∇·(Φ4πG ∂g∂t)−g4πG·∂g∂t,$
(17)

which, inserted into Eq. (14), transforms it into

$−ρV·g=∂∂t(−g28πG)+∇·[Φ(ρV−14πG ∂g∂t)].$
(18)

Equation (18) has the required conservation form; it is the looked-for counterpart in classical Newtonian gravity of Poynting's theorem in electromagnetism. Unlike Poynting's theorem, however, Eq. (18) is hardly ever mentioned, nor was it known to several colleagues I consulted. I have found only one advanced graduate level text where it is derived, but without any discussion of its role in local energy conservation and with the divergence term—the one feature not found in standard texts—removed immediately by integration over space.17

The gravitational energy density is

$uG=−g28πG,$
(19)

an intrinsically negative quantity; adding positive energy makes it less negative, i.e., decreases the magnitude of the gravitational field. The gravitational energy flux density is

$SG=Φ(ρV−14πG ∂g∂t),$
(20)

which is equal to gravitational potential $Φ$ multiplied by a quantity (Eq. (12)), which is the mass flux density $ρV$ made divergence-free by adding a term proportional to the time derivative of the gravitational field. This is precisely analogous to the electric current density made divergence-free by adding the displacement-current term. The technical term for this is advection of gravitational potential by the mass flow.18 An arbitrary constant added to $Φ$ does not change the divergence of $SG$ and, thus, has no effect on energy conservation.

Note that uG is analogous, except for the change of sign, to the energy density of the electric field in Poynting's theorem, and SG is analogous to the electric part of the much-discussed alternative form of the Poynting vector.16

The net gravitational energy flowing out of a volume enclosed by any equipotential surface $Φ=constant$ is given by the surface integral

$∫Φ=const.dA·SG=Φ∫Φ=const.dA·(ρV−14πG ∂g∂t)=0,$
(21)

which is zero by Eq. (12). The two terms inside the parentheses cancel, both representing, with opposite signs, the rate of change of the enclosed mass content: The second determined from its gravitational effect and the first from continuity of mass flow. Hence, the work done against gravity within the volume enclosed by a gravitational equipotential is balanced entirely by the change of gravitational energy contained within the volume, and there being no net flow of gravitational energy through the equipotential surface. The averaged gravitational energy flow is, thus, along the equipotential surfaces, and it is then a simple further step to reformulate the energy equation so that the local energy flow also is always along the equipotentials.

The divergence-free mass flow may be written as the curl of a vector $β$ defined by

$(ρV−14πG ∂g∂t)=−∇×β,$
(22)

which may be called the vector potential of the mass flow. The concept of vector potential is commonly introduced for magnetic fields, but any divergence-free field in three dimensions can be represented as the curl of a vector potential, just as any curl-free field can be represented as the gradient of a scalar potential. To evaluate $β$, take the curl of Eq. (22) (noting that $∇×g=0$) to obtain

$∇2β−∇(∇·β)=∇×ρV,$
(23)

which can be rewritten as

$∇2(β−∇Γ)=∇×ρV,$
(24)

where Γ is defined in terms of the divergence of $β$ by

$∇2Γ=∇·β.$
(25)

The solution for $β$ is

$β−∇Γ=14π∫d3r′ ∇′×ρ(r′)V(r′)|r−r′|=−14π∫d3r′ ρ(r′)V(r′)×(r−r′)|r−r′|3.$
(26)

Γ is arbitrary and for all practical purposes may be taken as zero, which is equivalent to setting $∇·β=0$.

Expression (22) may be inserted into the divergence term of Eq. (20), but an instructive alternative is to start with work done,

$−ρV·g=−g·[14πG ∂g∂t−∇×β]=−∂∂t(g28πG)−∇·(g×β)+β·∇×g.$
(27)

Given $∇×g=0$, this becomes the energy equation

$−ρV·g=∂∂t(−g28πG)−∇·(g×β),$
(28)

with energy flux density

$SG=−g×β.$
(29)

SG expressed by Eq. (29) is identical in the mathematical form to the Poynting vector; in fact, the entire derivation, Eqs. (22), (27), and (28), is analogous to the standard derivation of Poynting's theorem. Equation (29) has previously been derived by assuming that a velocity-dependent gravitational force implied by special relativity exists19 and is also discussed in Ref. 9.

Using Eqs. (9) and (16) (with $a=Φ$ and $b=∂Φ/∂t$), write

$Φ∂ρ∂t=ρ∂Φ∂t+∇·[g4πG ∂Φ∂t−Φ4πG ∂g∂t],$
(30)

then add $Φ∂ρ/∂t$ to both sides of the equation and divide by 2 to obtain

$Φ∂ρ∂t=∂∂t(12ρΦ)+∇·[g8πG ∂Φ∂t−Φ8πG ∂g∂t].$
(31)

Inserting this into Eq. (14) gives

$−ρV·g=∂∂t(12ρΦ)+∇·[Φ(ρV−18πG ∂g∂t)+g8πG ∂Φ∂t],$
(32)

which has the required conservation form. The gravitational energy density now is

$uG=12ρΦ.$
(33)

The factor 1/2 follows from the equations and need not be assumed a priori. The gravitational field energy flux density is

$SG=Φ(ρV−18πG ∂g∂t)+g8πG ∂Φ∂t,$
(34)

an expression involving both the potential and its time derivative, more complicated than Eq. (20) and without the latter's clear physical meaning. Adding an arbitrary constant to $Φ$ changes both uG and the divergence of $SG$, but the two changes compensate each other, so here also energy conservation is not affected. There is no counterpart of Eq. (21) because $∂Φ/∂t$ need not be constant on equipotential surfaces.

There are no specific analogs to Poynting's theorem; expressions of electromagnetic field energy density in terms of potentials are seldom used except in purely electrostatic or purely magnetostatic contexts.

In the three versions of the local energy conservation equation for Newtonian gravity derived here, Eqs. (18), (28), and (32), the field energy density uG appears in two forms: As field squared, Eq. (19), and as (1/2) density × potential, Eq. (33). The two, often derived by just visualizing the system assembled particle by particle, are well-covered in texts.20 They are related by the equation

$−g28πG=12ρΦ+∇·(Φg8πG),$
(35)

which follows from Eqs. (9) and (15). Locally, Eq. (35) is unchanged by adding an arbitrary constant to the potential, but when integrated over the entire system, the equation becomes

$−∫d3r g28πG=∫d3r 12ρΦ−12MΦ∞=∫d3r 12ρ(Φ−Φ∞),$
(36)

where M is the total mass and $Φ∞$ is the potential at infinity. The energy densities uG of the two expressions (19) and (33), thus, integrate to the same total energy UG only if $Φ∞=0$ is assumed, and even then they describe two quite different local distributions of energy.21

Which of these expressions for energy density and energy flux density are just mathematical functions constructed to ensure conservation of energy, and which—if any—have physical meaning? The objective criterion for physical meaning, other than quasi-philosophical arguments on what is “intuitively reasonable,” is that the expression be uniquely defined in terms of measurable physical parameters, which is a criterion satisfied by work done. In the terminology of Bridgman, a physical entity should have locally observable effects and, hence, should be detectable by suitable in situ instruments; quantities which can only be calculated from theory and/or measurements made elsewhere are to be considered mathematical constructs.

From this point of view, the uG of Eq. (19) is a locally measurable quantity and, thus, should be taken as the physical expression of gravitational energy density. The uG of Eq. (33) is a mathematical construct: It can only be calculated because a potential defined solely by position in a force field has no local signatures and, hence, cannot be locally measured. In Bridgman's example in Fig. 1, water in the upper reservoir has more potential energy than water in the lower reservoir, yet the properties of water in both reservoirs are the same. In other kinds of systems, where work is done not by displacing matter in a force field but by deforming a structure, e.g., a bent bow or stretched spring, increased potential energy is detectable as increased strain.

Gravitational energy flux density SG is a mathematical construct in all three versions of the energy equation, including the two that contain the physical expression Eq. (19) for uG: SG depends either on $Φ$ or on the vector potential $β$, which also has no local signatures. In Fig. 1, according to Bridgman, “[we] are driven… to associate with gravitating matter a store of energy which can be disclosed by no known measuring instrument.”11 The expressions are products of two terms: In Eq. (18), the not-locally measurable gravitational scalar potential is multiplied by the locally measurable mass flow, while in in Eq. (28), the locally measurable gravitational field is multiplied by the not-locally measurable vector potential of mass flow. Under what conditions field energy flux density can only be calculated although field energy density can be measured is further elucidated in Sec. VI.

A direct illustration of Eq. (21) is provided by the solar wind, the continuous outflow of plasma from the Sun that removes mass from the Sun at the rate of some $10−14$ solar masses per year.22–25 Although the work done to maintain the flow against solar gravity represents a potential energy input equal to nearly twice the kinetic energy of the flow itself, no gravitational potential energy is advected outward by the solar wind; instead, potential × mass outflow rate represents the rate at which the Sun's gravitational energy is changing as the result of mass loss. While it is not uncommon to speak of the solar wind as carrying gravitational potential energy, saying the same about solar photon radiation, the energy/mass equivalent of which represents a mass loss about a factor of four larger than mass removed by the solar wind, is rather unusual, although not entirely unknown.26

For the system sketched in Fig. 1, the field can be decomposed into the sum of the unchanging vertical field of the Earth $g0$ and the field of the water in the reservoirs $g1$,

$g=g0+g1, |g1|≪|g0|,$
(37)

with densities ρ0 (Earth) and ρ1 (water) and $Φ0, Φ1$ the corresponding gravitational potentials. The water reservoirs are treated as distinct from the Earth and above its surface.

To lowest order in $|g1|/|g0|$, Eq. (18) then becomes

$−ρ1V·g0=∂∂t(−g0·g14πG)+∇·[Φ0(ρ1V−14πG ∂g1∂t)],$
(38)

and the alternative Eq. (28) becomes

$−ρ1V·g0=∂∂t(−g0·g14πG)+∇·(g0×β).$
(39)

Only the interaction energy term $−2g0·g1$ appears, the self-energy term $g02$ being time-independent and $g12$ being negligibly small. Since $g0$ is vertical and downward, only the vertical component of $g1$ affects the energy density; an upward vertical $g1$ decreases the total $|g|$, hence increases the gravitational energy; a downward $g1$ does the reverse.

The energy expended by the work $−ρ1V·g0$ to pump the water goes into energy of the gravitational field, which is being modified by the changing $g1$ as the mass of water is transferred from lower to higher altitude. The path of energy flow from the pump to the perturbed field, as described by Eq. (38), begins with advection by the $ρ1VΦ0$ term, first upward within the connecting pipe and then horizontally into the upper reservoir; there follows the downward flow of gravitational energy represented by the $−(Φ0/4πG) ∂g1/∂t$ term, which presumably spreads horizontally over a wider area. This will roughly be a distance comparable in the order of magnitude either to the horizontal extent or to the vertical separation of the reservoirs, whichever is larger. Integrated over any one of the equipotential surfaces (approximately constant height, shown by dotted lines in Fig. 1) between the top and the bottom levels, the two terms cancel exactly: The upward advection of potential by flow within the pipe is balanced by the integrated downward flow of field energy everywhere produced by the time derivative of the changing $g1$. There is no net vertical flow of gravitational energy, although there may be localized up-and-down variations that average out to zero; the mechanical energy expended within each height range goes into changing the energy of the gravitational field within that height range.

In the alternative description of the energy flow by Eq. (39), the change of gravitational field energy is everywhere the same as above, but the energy flow pattern is different, proceeding horizontally outward from the pump and connecting pipe. $g0$ is vertical and essentially constant; $β$ circles around the pump and connecting pipe, and its magnitude decreases with increasing distance from them.

The total interaction energy of the reservoir-Earth system is a global quantity, requiring integration over the entire system. The change of the gravitational energy as water is pumped from lower to the upper reservoir (or vice versa), on the other hand, can be computed from the changes of $g1$, which are essentially local, becoming negligible at distances large compared to the horizontal extent of the reservoirs. In real life, the horizontal extent of a reservoir is likely to be much larger than its depth. The gravitational field $g1$ is then that of a thin flat plate: The vertical components point into the plate on both sides and decrease in magnitude only over vertical length scales comparable to the horizontal extent of the plate. If in addition the altitude difference of the reservoirs is much smaller than their horizontal extent, $g1$ can be represented by the following simple model.

Let $A=$ area of each reservoir, $h=$ altitude difference, $a=$ cross-sectional area of the connecting pipe, and $V=$ the speed of water in the pipe. Designate the amount of water mass in the two reservoirs as $(m)lower, (m)upper$, respectively, subject to the constraint,

$(m)lower+(m)upper=m1,$
(40)

where m1 is the constant total mass. If water is being pumped from the lower to the upper reservoir,

$ρ1Va=ddt(m)upper=−ddt(m)lower.$
(41)

Over the height range between dotted lines b–b′ and c–c′ in Fig. 1, the vertical components of $g1$ are nearly constant and point into the respective reservoirs with values

$(g1)lower=−(2πGA)(m)lower, (g1)upper=(2πGA)(m)upper$
(42)

(positive = upward). The nearly uniform field of each reservoir occupies approximately a volume hA within the assumed height range. The change of gravitational energy integrated over the volume is

$ddt∫b−b′c−c′ d3r(−g0·g14πG)≃g0h2ddt[(m)upper−(m)lower],$
(43)

or, on evaluating the right side from Eq. (41),

$ddt∫b−b′c−c′ d3r(−g0·g14πG)≃ρ1Vg0ha.$
(44)

Thus, the integrated energy change equals total work done.

In regions above d–d′ and below a–a′ (Earth's surface), where negligible work is done, significant non-uniform fields exist, but their contribution to the change of energy is zero when integrated horizontally, as discussed above.

For the system sketched in Fig. 1, Eq. (32) can be written, again to lowest order in $|g1|/|g0|$, as

$−ρ1V·g0=∂∂t(12ρ1Φ0+12ρ0Φ1)+∇·(ρ1VΦ0)+∇·[18πG (g0∂Φ1∂t−Φ0∂g1∂t)].$
(45)

Unlike the case discussed in Subsection V B above, there is now a net vertical flow of gravitational energy. Integrate Eq. (45) separately over the volumes above and below the surface of the Earth,

$−∫surface∞ d3r ρ1V·g0=ddt[∫surface∞ d3r 12ρ1Φ0]−S, above$
(46)
$0≃ddt[∫0surfaced3r 12ρ0Φ1]+S, below$
(47)

where

$S=∫dA r̂·[18πG (g0∂Φ1∂t−Φ0∂g1∂t)]≃∫dA r̂·g08πG ∂Φ1∂t$
(48)

is the integral of the gravitational energy flux density over the interface surface between above and below (i.e., Earth's surface), representing the total rate of energy transfer from above to below. In the second expression, time independence of the surface integral of $g1$, proportional to the total external mass m1, has been invoked.

By the reciprocity relation of the  Appendix, the two integrals within brackets in Eqs. (46) and (47) are equal to each other at all times. As a consequence, adding and subtracting these two equations and dividing by 2 give the relations

$−12∫surface∞ d3r ρ1V·g0=ddt[∫surface∞ d3r 12ρ1Φ0]=ddt[∫0surfaced3r 12ρ0Φ1],$
(49)
$−12∫surface∞ d3r ρ1V·g0=−S.$
(50)

When expressed in terms of potentials, energy supplied by the work is partitioned: One half changes the potential energy stored in the reservoirs, and the other half flows down to change the potential energy stored in the Earth.

Poynting's theorem is arguably the best-known of all the local field energy conservation equations. In its standard form of Eq. (7) or (8), it has electromagnetic field energy density expressed in terms of fields and energy flux density given by the Poynting vector; an alternate form of energy flux density in terms of potentials is also well known.16

There are similarities and differences between both forms of Poynting's theorem and the corresponding forms of the Newtonian energy equation. Aside from differently defined quantities, units, and change of sign, one obvious difference is the presence of two locally measurable force fields E and B in electromagnetism, in contrast to the one field g of gravity. As shown in Sec. III B, however, the Newtonian energy equation can be expressed in the form of Eq. (28), mathematically identical with Poynting's theorem, which also contains two fields g and $β$, the only difference being that $β$ appears in the energy flux density term (the counterpart of the Poynting vector) but not in the energy density term. To understand why this is so, it is instructive to rederive Poynting's theorem, by analogy with the derivation of the Newtonian Eq. (28), taking charge conservation as the starting point,

$∇·(J+14π ∂E∂t)=0,$
(51)

and not introducing the magnetic field until it becomes necessary. Equation (51) implies that the divergence-free electric current density can be written as the curl of a vector potential $β*$,

$(J+14π ∂E∂t)=∇×β*.$
(52)

Inserting J given by Eq. (52) into work done then gives

$−E·J=E·[14π ∂E∂t−∇×β*]=∂∂t(E28π)+∇·(E×β*)−β*·∇×E$
(53)

(Gaussian units; for SI, replace $1/4π$ by ϵ0). If we assume the electrostatic approximation $∇×E=0$, Eq. (53) becomes

$−E·J=∂∂t(E28π)+∇·(E×β*),$
(54)

a conservation equation precisely identical in form to the Newtonian Eq. (28), including a form of the Poynting vector. If Eq. (53) is to be put into conservation form without assuming the electrostatic approximation, $∇×E$ must satisfy the equation

$∇×E=−λ ∂β*∂t,$
(55)

where λ is a constant; the last term of Eq. (53) then becomes the time derivative of an added energy density $(λ/2)|β*|2$. Equation (55) can now be identified with Faraday's law, thereby introducing explicitly the physical magnetic field B and establishing its relation to the mathematical construct $β*$,

$β*=c4π B Gaussian units,$
(56)
$β*=Bμ0 SI units$
(57)

($λ=c2/4π, λ=μ0$ for Gaussian and SI units, respectively).

What makes the above clear is that, as far as conservation of energy is concerned, the difference between Newtonian gravity and electromagnetism results from the assumption $∇×g=0$ in contrast to $∇×E≠0$: The magnetic field appears as an added energy density term in Poynting's theorem (despite the fact that magnetic force does no work) because of Faraday's law, but there is no corresponding added term in the Newtonian equation. The two expressions for energy flux density, on the other hand, can be put into identical forms except for a sign difference,

$SEM=E×β* electromagnetic,$
(58)
$SG=−g×β gravitational,$
(59)

where $β*, β$ are the vector-potential representations of the divergence-free current density and the divergence-free mass flux density, respectively. $β$ is related to $ρV$ by Eq. (26), and $β*$ is related to J by a similar equation with the Laplacian replaced by the speed-of-light wave operator. Both the energy flux densities, Poynting vector and its Newtonian counterpart, are determined by the values of $β*$ and $β$, independently of whether or not these are related to locally measurable quantities (and if so, how).

These mathematical constructs can, in fact, be related to physical quantities. The electromagnetic $β*$ is proportional to the magnetic field, according to Eqs. (56) and (57). The gravitational $β$ has no observable physical effects within the strictly Newtonian theory, but an additional acceleration proportional to the particle velocity v,

$−(v/c)×h,$
(60)

appears in the linear (post-Newtonian) approximation to general relativity.27 The “gravomagnetic field” h is related to $β$ by

$β=c4πG (h4),$
(61)

similar to the electromagnetic Eq. (56) relating B to $β*$ except that h is divided by 4, which is a general relativistic effect: 2 from gravitational time dilation plus 2 from space curvature.28 I have put “gravomagnetic field” in quotation marks as a reminder that general relativity treats gravity in a fundamentally different way than Newtonian theory. The analogy between B and h is mathematical: Both are measurable (in principle!) by observing the acceleration of a moving test particle, but B is a physical force field acting on the particle while h is the consequence of modified space-time geometry.

The physical significance of these relations is readily apparent for the electromagnetic case in Gaussian units, Eq. (56), as well as for the gravitational case, Eq. (61): The mathematical construct is proportional to the physical quantity multiplied by the speed of light. This suggests that what confers independent physical reality to flow of field energy is finite propagation speed, as alluded to in Sec. I. The sources of the field (charge or mass) change on a time scale set by the flow of the source (current or mass flow); as long as the field configuration adjusts itself to the changing sources instantaneously as assumed in electrostatics and in Newtonian gravity, field energy distribution and source distribution are both changing on the same time scale; hence, the field energy flow as a physical process distinct from the source flow is not observable. When the adjustment of the field to changes in its sources is no longer instantaneous, however, mathematical terms constructed to express balance between energy supply and storage become identifiable as physical locally measurable fields, varying on time scales that can differ from those of their sources. Consistent with this interpretation, for a given energy flow rate, these fields are proportional to $1/c$. When c can be regarded as effectively infinite relative to spatial and temporal scales, the physical field approaches negligible values while the mathematical construct remains finite.

Whether the above interpretation could be inferred from Eq. (57) in SI units instead of Eq. (56) in Gaussian units is not clear. It has been repeatedly pointed out that SI units treat electric and magnetic fields asymmetrically, thereby obscuring the fundamental symmetry between them required by special relativity.29

1. A local energy conservation equation for Newtonian gravity, which plays the same role as Poynting's theorem for electromagnetism, has been derived in three different forms: Eqs. (18), (28), and (32). All three include the divergence of a gravitational energy flux density; this term, analogous to the divergence of the Poynting vector in electromagnetism, has been left out in almost all previous discussions of Newtonian gravity with the exception of some obscure and highly specialized texts.

2. The gravitational energy flux density in Eq. (28) is expressed in a mathematical form identical to that of the Poynting vector. This expression, derived previously only by invoking first-order relativistic terms, is here purely obtained from the field equations of Newtonian gravity, confirming that Newtonian theory is logically complete by itself and does not require reference to relativistic effects.

3. Conversely, a local energy conservation equation for electromagnetism can be purely derived from electrostatic equations (assuming $∇×E=0$), without introducing the magnetic field. The resulting equation (54), analogous to the gravitational equation (28), is a truncated Poynting's theorem: It contains no magnetic energy term but does contain the divergence of a field energy flux density, identical to the usual Poynting vector except for replacement of the magnetic field by a mathematical construct, the curl of which equals the current density without the proportionality constant ($4π/c$ in Gaussian or μ0 in SI units). I am not aware of any previous publication of this “electrostatic” Poynting's theorem.

4. The energy density of the gravitational field in Eqs. (18) and (28) is the well-known expression in terms of the local gravitational field and, thus, can be identified with a locally measurable physical quantity. The energy density in Eq. (32), the equally well-known expression in terms of density and potential, is calculable but not locally measurable. Both expressions have been derived here rigorously from the field equations.

5. The gravitational energy flux density in all three forms of the Newtonian energy equation is calculable but is not locally measurable. Comparison with Poynting's theorem and with the post-Newtonian approximation to general relativity suggests that field energy flux density becomes locally observable when the finite speed of light is taken into account.

The author's search for a Newtonian analog to Poynting's theorem was motivated by conversations with the late Professor Stanislaw Olbert (M.I.T.). The author is grateful to George Siscoe, Karl Schindler, Ramón López, Alex Dessler, Forrest Mozer, and the late F. Curt Michel for enlightening discussions and to the referees for useful comments.

The author has no conflict of interest.

Consider an extended mass m0 filling a finite closed volume, plus another extended mass m1 outside the volume. The gravitational field can be expressed as the sum

$g=g0+g1,$
(A1)

where $g0, g1$ and the corresponding gravitational potentials $Φ0, Φ1$ are related by Poisson's equation (9) to the densities ρ0, ρ1, respectively. Since the energy of the gravitational field is expressed, in all its forms, as a product of two factors, the energy UG of the entire system can be written as the sum

$UG=Uself,0+Uself,1+Uint,$
(A2)

where $Uself,0, Uself,1$ are the self-energies of the two masses (each in isolation) and $Uint$ is their interaction energy. With the field energy in terms of potentials, Eq. (33), the energies are given by the integrals

$Uself,0=12∫d3r ρ0Φ0, Uself,1=12∫d3r ρ1Φ1,$
(A3)
$Uint=12∫d3r ρ1Φ0+12∫d3r ρ0Φ1,$
(A4)

which can be formally taken over all space as the integrands are automatically non-zero only over the volumes of the two masses.

##### 1. Reciprocity relation between exterior and interior potential energies

The interaction energy Eq. (A4) is the sum of two integrals, which can be related to the help of Green's identity (16) rewritten as

$ρ1Φ0−ρ0Φ1=∇·[14πG (Φ1g0−Φ0g1)].$
(A5)

Integrating Eq. (A5) over all space and remembering that

$m=∫d3r ρ=−∫dA r̂·g4πG$
(A6)

yields

$∫d3r ρ1Φ0−∫d3r ρ0Φ1=Φ∞(m1−m0)$
(A7)

or

$∫d3r ρ1(Φ0−Φ∞)=∫d3r ρ0(Φ1−Φ∞),$
(A8)
$∫d3r ρ1Φ0=∫d3r ρ0Φ1 if Φ∞=0.$
(A9)

This is the reciprocity relation: The potential energy of mass m1 in the field of mass m0 equals the potential energy of mass m0 in the field of mass m1. With the two integrals in Eq. (A4) equal, the interaction energy $Uint$ is partitioned between the two masses in the ratio 1/2 to 1/2.

If, within either volume, the distribution of density undergoes a change ($ρ→ρ′$) without the change of total mass m, the changes within both volumes are still related by Eq. (A7),

$∫d3r ρ1′Φ0′−∫d3r ρ0′Φ1′=Φ∞(m1−m0).$
(A10)

Subtracting Eq. (A7) from Eq. (A10), rearranging terms, and defining $Δ(ρΦ)≡ρ′Φ′−ρΦ$ give

$∫d3r Δ(ρ1Φ0)=∫d3r Δ(ρ0Φ1),$
(A11)

for any value of $Φ∞$. The reciprocity relation, thus, holds also for changes of potential energy without the need to assume $Φ∞=0$; the change of interaction energy $ΔUint$ is partitioned equally between the two masses.

##### 2. Interaction energy in terms of fields

With the field energy in terms of fields, Eq. (19), the energies are given by the integrals over all space,

$Uself,0=∫d3r (−g028πG), Uself,1=∫d3r (−g128πG),$
(A12)
$Uint=∫d3r (−g0·g14πG).$
(A13)

$Uint$ is now a single integral over all space, symmetrically dependent on both $g1$ and $g0$; the integrand is non-zero everywhere and locally can be positive or negative; the integrated $Uint$, however, is always negative. The separation of contributions from each mass is not possible, but an interesting albeit non-unique association can be made. Write Green's identity (15) in two equivalent versions:

$−g0·g1=∇·(Φ0g1)−Φ0∇·g1,$
(A14)
$−g1·g0=∇·(Φ1g0)−Φ1∇·g0.$
(A15)

Select the equipotential $Φ0$ of the field of mass m0 enclosing the volume that includes the mass m0 and just touches but does not include the mass m1. Integrating Eq. (A14) over the volume interior to the equipotential surface $Φ0$ gives

$∫intd3r (−g0·g14πG)=−Φ04πG∫dA·g1=0.$
(A16)

The surface integral is zero because the surface does not enclose the mass m1. The interaction energy density is not zero inside the volume, however, but takes both negative and positive values which cancel when summed; $g1·g1$ is mostly negative in the half-volume closer to m1 and positive in the other half. With the integral over the interior volume equal to zero, the integral over the volume exterior to the equipotential, which does include the mass m1, equals the integral over all space, which by Eq. (A13) equals the interaction energy,

$∫extd3r (−g0·g14πG)=Uint.$
(A17)

Selecting the equipotential $Φ1$ instead of the field of mass m1 enclosing the volume that includes the mass m1 and just touches but does not include the mass m0, then integrating Eq. (A15) over the volumes interior and exterior to $Φ1$, respectively, gives identical results with the masses m1 and m0 interchanged.

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