Foundations of Physics
(2023) 53:15
https://doi.org/10.1007/s10701-022-00660-z
Energy in Newtonian Gravity
Ingemar Bengtsson1 · Tobias Eklund1,2
Received: 23 December 2021 / Accepted: 30 November 2022
© The Author(s) 2022
Abstract
In Newtonian gravity it is a moot question whether energy should be localized in the
field or inside matter. An argument from relativity suggests a compromise in which
the contribution from the field in vacuum is positive definite. We show that the same
compromise is implied by Noether’s theorem applied to a variational principle for
perfect fluids, if we assume Dirichlet boundary conditions on the potential. We then
analyse a thought experiment due to Bondi and McCrea that gives a clean example
of inductive energy transfer by gravity. Some history of the problem is included.
Keywords Newtonian gravity · Localisation of energy · Tidal forces
1 Introduction
How is a local energy density to be defined in Newtonian gravity? This is a moot
question. Traditionally, the two main contenders for a definition are that of Maxwell
[1],
E� = −
1
𝜕 Φ𝜕 Φ ,
8𝜋G i i
(1)
and an alternative where the energy is localized within the matter,
1
E�� = − 𝜌Φ .
2
(2)
We will refer to the latter as Bondi’s energy density, because it was championed by
him [2, 3]. Our conventions for the gravitational interaction are set by the equations
that connect the gravitational potential Φ to the mass density of matter 𝜌,
* Ingemar Bengtsson
ibeng@fysik.su.se
1
Stockholms Universitet, AlbaNova, Fysikum, Stockholm 106 91, Sverige
2
Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany
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∇2 Φ = −4𝜋G𝜌
Foundations of Physics
⇔
Φ(x, t) = G
(2023) 53:15
𝜌(x� , t) 3 �
d x.
∫ |x − x� |
(3)
We have adopted the convention that the potential Φ is positive. At some points we
may set the constant G to 1. The formula using the Green function assumes that the
matter density has compact support, so that we deal with an isolated system.
The total energy is obtained by integrating the energy density over all space,
and comes out the same regardless of whether we use E′ or E′′ . There is an analogous issue in electrostatics. Maxwell, who was thinking of force fields as emergent from a medium, insisted that the choice is important:
“I wish to be understood literally. All energy is the same as the mechanical
energy, whether it exists in the form of motion or in that of elasticity ...The only
question is, Where does it reside? ... On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well
as in those bodies themselves.” [1]
But Maxwell faced a problem with gravity. The energy density is negative
definite while (in his view) energy is “essentially positive”. As a way out he suggested that a constant should be added to E′ to make it everywhere positive. But
this would mean that the energy density of the medium must be huge where the
gravitational field is weak. He concluded:
“As I am unable to understand in what way a medium can possess such properties, I cannot go any further in this direction in searching for the cause of gravitation.” [1]
It should be noticed that the notion of energy conservation was by no means
uncontroversial at the time. Herschel regarded it as a verbal trick [4]. Much later,
Mason and Weaver [5] argued that energy is a function of the configuration of the
system as a whole, and that it is no more sensible to inquire about the location of
energy than to declare that the beauty of a painting is distributed over the canvas
in a specified manner.
Who is right: Maxwell, Bondi, or Mason and Weaver? For electromagnetism
the question is often regarded as resolved (in favour of Maxwell) by the relativistic theory, and in particular by the way that the electromagnetic field couples to
gravity. For gravity there is no external arbiter to make the decision.
In general relativity the total energy of an isolated system is well understood,
but the localisation of energy is a tangled question indeed. The energy at a point
can be argued away using the equivalence principle, but there are proposals for
the energy located within some chosen closed surface. Some of these proposals,
notably those of Hawking [6] and Penrose [7], suggest that the energy of a black
hole is located within its event horizon. This is perhaps reminiscent of Bondi’s
expression in the Newtonian case. Others, notably Lynden-Bell and Katz [8],
have proposed expressions that are closer in spirit to that of Maxwell. To support their case Lynden-Bell and his coworkers considered a static gravitational
field coupled to a perfect fluid. We then lose very little by assuming spherical
symmetry as well, so that the formulas that follow should be familiar to most
readers. To obtain a conserved current from the relativistic stress-energy tensor Tab we need to contract it with a timelike Killing vector 𝜉 b , and the energy
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15
density gains an extra factor coming from the norm of the Killing vector field.
Let ta be a timelike unit vector and set
Tab = (𝜌 + p)ta tb + pgab
𝜉a =
√
−gtt ta ,
(4)
−gtt = 1 − 2Φ.
We can build an energy density 𝜇 and take its Newtonian limit,
√
𝜇 = −Ttb 𝜉 b = 𝜌 1 − 2Φ ≈ 𝜌(1 − Φ) = 𝜌 + 2E�� .
(5)
(6)
Maxwell’s very large constant appears here in the guise of the mass density 𝜌, but
the binding energy is twice as large as he may have expected. To make the total
energy come out as being equal to the mass plus the Newtonian energy we add a
term coming from the gravitational field itself. We can then define a ‘true’ Newtonian energy density E through [8, 9]
𝜌 + E = 𝜌 + 2E�� − E� = 𝜌(1 − Φ) +
1
𝜕 Φ𝜕 Φ.
8𝜋 i i
(7)
Maxwell’s sign problem has evaporated in the sense that the energy density is now
positive definite outside matter, although the argument rests on the assumption that
the gravitational field is static.
This argument did not lead to a consensus. Frauendiener and Szabados [10]
use post-Newtonian corrections to Newton’s theory to argue that the energy density integrated over large volumes, enclosing all the matter, should be a monotonically decreasing function of the volume. So the question is indeed moot.
As far as we know the expression E = 2E�� − E� for the energy density was first
advocated by Ohanian [11]. The question continues to attract interest [12]. Our
first aim here is to see how one can argue in favour of the energy density E from
within the Newtonian theory itself, at the same time dropping the assumption
of static fields. We will do this by appealing to Noether’s theorem. In view of
its hundreth anniversary this theorem has attracted interest from philosophers
of science recently [13], and indeed Dewar and Weatherall used it to study the
energy concept in Newtonian gravity [14]. However, since they used an external matter source they were unable to address the question that we consider in
Sect. 2. We add that other aspects of their paper have given rise to illuminating
discussions [15]. In particular the Newton–Cartan formulation of the theory has
been a subject for these discussions. This is an interesting topic about which we
have nothing to say.
In Sect. 3 we go on to consider an interesting example of energy transport
in Newtonian gravity, in rather more detail than was offered in the original
paper by Bondi and McCrea [16]. In Sect. 4 we draw attention to the fact that
the energy density proposed in Sect. 2 plays no role in the concrete setting of
Sect. 3. The question then becomes one of the pragmatic advantages of the various definitions.
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2 An Energy Density from Noether’s Theorem
In Newtonian gravity we need matter to provide dynamics. A fluid described by
a mass density 𝜌 and a velocity field vi is appropriate. These variables obey mass
conservation
𝜕t 𝜌 + 𝜕i (𝜌vi ) = 0
(8)
as well as the Euler equation
𝜌
d
v = 𝜌(𝜕t vi + vj 𝜕j vi ) = 𝜕j 𝜏ij + 𝜌𝜕i Φ.
dt i
(9)
Here 𝜏ij is the stress tensor, and the last term is the gravitational body force. In the
spirit of Maxwell the latter can be regarded as being due to gravitational stress, but
for this we refer to Synge [17]. For simplicity we will set 𝜏ij to zero and assume that
the velocity field is irrotational. Thus our matter consists of irrotational gravitating
dust, but we comment briefly on more general cases at the end. That the velocity
field is irrotational means that there exists a velocity potential 𝜆,
𝜕i vj − 𝜕j vi = 0
⇔
vi = −𝜕i 𝜆.
(10)
The potential is essential in order to find an action integral from which Euler’s equations can be derived [18–20].
We begin by recalling how the variational principle works. In the action the
velocity potential appears as a Lagrange multiplier imposing conservation of
mass:
[
]
(
)
1
1
S0 [𝜌, vi , 𝜆, Φ] =
𝜌vi vi − 𝜆 𝜕t 𝜌 + 𝜕i (𝜌vi ) + 𝜌Φ −
𝜕i Φ𝜕i Φ d4 x. (11)
∫ 2
8𝜋
Varying the action with respect to the velocity field vi we recover Eq. (10), which is
an algebraic equation for vi that can be inserted in the action. The result is
[
]
1
1
S1 [𝜌, 𝜆, Φ] =
𝜌𝜕t 𝜆 − 𝜌𝜕i 𝜆𝜕i 𝜆 + 𝜌Φ −
𝜕i Φ𝜕i Φ d4 x .
(12)
∫
2
8𝜋
A total time derivative was added. This amounts to a canonical transformation in the
matter sector, and does not affect our later arguments. Total space derivatives are
important and will be discussed soon. Variation with respect to Φ returns Poisson’s
equation (3), and variation with respect to 𝜌 gives
1
𝜕t 𝜆 − 𝜕i 𝜆𝜕i 𝜆 + Φ = 0.
2
(13)
Taking the gradient of this equation and making use of eq. (10) yields
−𝜕t 𝜕i 𝜆 + 𝜕j 𝜆𝜕j 𝜕i 𝜆 = 𝜕i Φ
⇒
𝜕t vi + vj 𝜕j vi = 𝜕i Φ.
This is the equation of motion for irrotational dust.
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The velocity potential 𝜆 has no direct physical interpretation since it is defined
only up to a constant. In a Galilei invariant model we insist on the mass superselection rule, that is we insist that all observables Poisson commute with the total
mass [20]
M=
∫
𝜌 d3 x.
(15)
Clearly {M, 𝜆} = 1, so indeed 𝜆 is not an observable while its gradient is.
Before we apply Noether’s theorem we must address the issue of possible surface terms that can be added to the action. In infinite space they play no role in
deriving the equations of motion, but we will consider a finite spacetime region
V with an enclosing surface placed in vacuum [21]. The action amended with a
surface term is
a
𝜕 (Φ𝜕i Φ)d4 x
4𝜋 ∫ i
]
[
1
a
1
𝜕i Φ𝜕i Φ +
𝜕i (Φ𝜕i Φ) d4 x.
𝜌𝜕t 𝜆 − 𝜌𝜕i 𝜆𝜕i 𝜆 + 𝜌Φ −
=
∫
2
8𝜋
4𝜋
(16)
Here a is a parameter to be fixed. Assuming that the field equations hold we find that
(
[
)]
a−1
a
𝛿S =
𝜕t (𝛿𝜆𝜌) + 𝜕i 𝜌vi 𝛿𝜆 +
𝜕i Φ𝛿Φ +
Φ𝜕i 𝛿Φ d4 x.
(17)
∫V
4𝜋
4𝜋
S[𝜌, 𝜆, Φ] = S1 [𝜌, 𝜆, Φ] +
The surface terms must vanish if the action is to be used to derive the field equations
in the bounded region. The total time derivative is not relevant for our problem, but
the total divergence is. We have assumed that 𝜌 = 0 on the spatial part of the boundary, but we still have to decide what boundary conditions to impose on Φ. This is the
question of how we control the system [21], and an answer will fix the parameter a.
The two most obvious choices give
Dirichlet
⇒
𝛿Φbdry = 0
⇒ a=0
Neumann ⇒ ni 𝛿𝜕i Φbdry = 0 ⇒ a = 1,
(18)
where the vector ni is normal to the boundary. In electrostatics it would be a straightforward task for the experimentalist to impose Dirichlet conditions on the electrostatic potential. For gravity the question is not so easily settled, but Dirichlet conditions do seem to be a natural choice. We will return to discuss this issue once we
have discussed a concrete example of gravitational energy transport. Meanwhile we
simply assume that a suitable choice has been made, so that the parameter a is fixed.
We are now ready to apply Noether’s theorem. Consider a rigid time translation,
𝛿𝜆 = 𝜖𝜕t 𝜆
𝛿𝜌 = 𝜖𝜕t 𝜌
𝛿Φ = 𝜖𝜕t Φ
⇒
𝛿L = 𝜖𝜕t L,
(19)
where L is the integrand of the action integral. Noether’s theorem follows from the
observation that for these variations we have the alternative expression
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Foundations of Physics
𝛿S =
∫
(2023) 53:15
𝜕t (𝜖L) d4 x.
The equality of the two expressions for 𝛿S implies the local conservation law
)
(
1 − 2a
1
𝜕i Φ𝜕i Φ
𝜕t 𝜌𝜕i 𝜆𝜕i 𝜆 + (a − 1)𝜌Φ +
2
8𝜋
(
)
a−1
a
+ 𝜕i 𝜌vi 𝜕t 𝜆 +
𝜕t Φ𝜕i Φ +
Φ𝜕i 𝜕t Φ = 0.
4𝜋
4𝜋
(20)
(21)
The energy density (including the kinetic energy of matter) can be read off from the
first term. If Dirichlet conditions are assumed we set a = 0 and find that the energy
density is
E=
1
1
1
𝜌v v − 𝜌Φ +
𝜕 Φ𝜕 Φ = 𝜌vi vi − E� + 2E�� .
2 i i
8𝜋 i i
2
(22)
This is not a decision between Maxwell and Bondi, it is a compromise in the sense
that energy is partly localized within matter and partly spread throughout the vacuum. It bears the mark of a good compromise since it agrees with the answer arrived
at by Lynden-Bell and Katz, as in Eq. (7). And the point we wanted to make is precisely that with suitable assumptions this answer can be derived from within the
Newtonian theory itself.
However, had we chosen to impose Neumann rather than Dirichlet boundary conditions we would have ended up with Maxwell’s expression for the energy density.
That Noether’s theorem leads to different expressions for the energy depending on
the boundary conditions we impose should, in fact, not be surprising. The situation
is interestingly analogous to how internal and free energy appear in thermodynamics, depending on how the system is controlled [21].
The second term in the local conservation law (21) gives the local energy flux.
We will return to it in Sect. 4 below. Meanwhile we observe that if we use the field
equations to clear the conservation law of time derivatives we obtain
)
(
)
(
1
1
𝜕t 𝜌vi vi + 𝜕i vi 𝜌vj vj = 𝜌vi 𝜕i Φ.
(23)
2
2
This is of course indisputable—the local kinetic energy changes due to work done
by the gravitational field—but any gravitational contribution to the energy density
has disappeared.
We have given the argument for irrotational dust. The entire argument clearly
goes through if we add a pressure term to the equations. The restriction to irrotational flow can be dropped too, if we make use of Clebsch potentials. An economical choice is that due to Seliger and Whitham [18],
vi = −𝜕i 𝜆 − 𝛼𝜕i 𝛽.
(24)
An additional pair of Clebsch potentials is needed to handle general flows globally
[22]. Applying Noether’s theorem to the action proposed by Seliger and Whitham
results in the same expressions for the local energy density and the local energy
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15
transport as the ones we just derived, once they have been expressed in terms of 𝜌,
vi , and Φ. For this reason we do not give the details here.
3 Tweedledum and Tweedledee
The arguments in Sect. 2 were rather formal. It is interesting to see how they fare in
a concrete problem. We do know that energy is being transported by gravity within
the solar system, in a way that is well described by Newtonian theory. As an example, tidal friction within the Earth–Moon system causes the Moon to move away
from the Earth. But in this case it is not easy to pinpoint where the energy ends up.
A more dramatic example is provided by the tidal heating of Io, one of the moons
of Jupiter [23]. A simpler example is called for here. Bondi and McCrea invented
a thought experiment in which tidal forces give rise to a net energy transport even
though the gravitational field returns to its initial state after some energy has been
transmitted [16]. The experiment concerns two mutually gravitating bodies in elliptical orbits around each other. Bondi later named them Tweedledum and Tweedledee [2], although in the original paper they were referred to as the receiver (R) and
the transmitter (T). They have spherical outlines, but their mass distributions can be
changed between prolate and oblate with the axial direction orthogonal to the orbital
plane. If they both turn oblate the gravitational attraction between them grows. If
one of them turns oblate and the other prolate the attraction can be kept constant,
and this is the key to the whole idea since it will allow them to stay on their elliptical orbits throughout the duration of the experiment. The changes of shape are controlled by some machinery powered by batteries external to the system that we will
describe, which goes to say that we will study an open system. Tidal forces tend to
make a body oblate, and when this happens work is done on the body. Conversely,
work is done by the body when it turns prolate. This is how transmission of energy
can occur. The question of how energy is stored in the gravitational field is avoided
because, as far as the gravitational field is concerned, the process is cyclic and we
will compute the energy transmitted during a full cycle.
The original paper is very brief, and we feel that it may be useful to tell the story
using equations. Full calculational details are given elsewhere [24]. The twins are
modelled as spheres with radii r0 and total mass M in both cases. Their mass densities have time dependent quadrupole moments QR = QR (t) and QT = QT (t) that can
be freely prescribed. In coordinate systems with origos at the centres of the bodies
{
3 M
35 QR (t) 2
+ 4𝜋
r P2 (cos 𝜃) , r < r0
4𝜋 r03
r07
𝜌R =
(25)
0
, r > r0
{
𝜌T =
3 M
4𝜋 r03
0
+
35 QT (t) 2
r P2 (cos 𝜃)
4𝜋 r07
, r < r0
, r > r0
(26)
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Foundations of Physics
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where P2 is a Legendre polynomial and 𝜃 is the angle against the normal to the
orbital plane. A body is prolate if its quadrupole moment is positive.
For the calculations to follow we need some formulas for the spherical harmonics
Y𝓁,m. We recall that
√
4𝜋
(27)
Y (x),
P𝓁 (x) =
2𝓁 + 1 𝓁,0
where the functions depend only on the direction of the vector in the argument. The
direction cosine of the vector is taken relative an axis orthogonal to the orbital plane.
For vectors x, y with lengths related by y < x the translation theorem states that
Y𝓁,m (x − y)
|x − y|𝓁+1
=
∑∑
𝓁�
m�
y𝓁
�
m,m
C𝓁,𝓁
�
�
x𝓁+𝓁� +1
Y𝓁+𝓁� ,m+m� (x)Y𝓁∗� ,m� (y)
(28)
′
m,m
where the expression for the coefficients C𝓁,𝓁
′ is somewhat unwieldy. Because we
are going to integrate against the monopole-quadrupole massdensities we need only
two terms in each of two special cases,
y2
1
1
= + ⋯ + 3 P2 (x)P2 (y) + ⋯
|x − y| x
x
(29)
P2 (x − y) P2 (x)
6y2
= 3 + ⋯ + 5 P4 (x)P2 (y) + … .
3
|x − y|
x
x
(30)
For a proof
see van Gelderen, who has an easily corrected misprint in his expression
m,m′
for C𝓁,𝓁
′ [25].
Using Eq. (29) we can now calculate the gravitational potential outside the receiver
as
ΦR (x, t) = G
𝜌R (x� , t) 3 � GM GQR
dx =
+ 3 P2 (x).
∫ |x − x� |
r
r
(31)
There is a similar formula for ΦT. We recall that, in Newtonian gravity, the total
self-force on a body vanishes [26, 27], and we calculate the total work needed to
place the receiver at a distance D from the transmitter. Using a vector D of length D
we need to calculate
VR = −
∫
𝜌R ΦT d3 x = −
∫
𝜌R (x)ΦT (x + D) d3 x.
(32)
Appealing to Eq. (30) we find that
VR = −
GM 2 GM(QR + QT ) 9GQR QT
.
−
+
D
2D2
4D5
(33)
By the terms of the agreement between the twins Tweedledum can choose the quadrupole moment QR = QR (t) at will, but Tweedledee must then adapt the function
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15
QT = QT (t) in such a way that the gravitational force between them stays the same as
in the two-body problem for spherical bodies. Thus we impose
𝜕VR
GM 2
GM 2 3GM(QR + QT ) 45GQR QT
.
−
+
=
=
𝜕D
D2
2D4
4D6
D2
(34)
The solution is
QT = −
2MD2 QR
.
2MD2 − 15QR
(35)
For later use we also record the differential form of the constraint,
(15QT − 2MD2 )dQR + (15QR − 2MD2 )dQT − 4MD(QR + QT )dD = 0.
(36)
We assumed that 2MD2 > 15QR . With this precise choice for the quadrupole
moments the orbits of the twins are ellipses with a common focus at the midpoint
of the line between them, and the mutual distance D = D(t) has a specified time
dependence. It is a periodic function, and we assume that Tweedledum chooses a QR
with the same periodicity.
We now wish to calculate the rate of work Ẇ R done on the receiver as his
quadrupole moment is changing. Bondi and McCrea use an elegant shortcut for
this purpose, but it is interesting to calculate it using field theory. For clarity we
decide that Ẇ R is to be calculated in an inertial system where the centre of mass
of the receiver is momentarily at rest, while Ẇ T is calculated in a system where
the transmitter is momentarily at rest.
From Eq. (9) we see that the rate of gravitational work is
Ẇ R =
∫
𝜌R vi 𝜕i ΦT d3 x.
(37)
Making use of the conservation of mass, Eq. (8), this can be rewritten in two useful
ways. Either as
Ẇ R =
∫
[
]
𝜕i (𝜌R vi ΦT ) − ΦT 𝜕i (𝜌R vi ) d3 x =
∫
ΦT 𝜕t 𝜌R d3 x
where a surface term was discarded, or as
(
)
d
d
Ẇ R =
𝜌R
ΦT − 𝜕t ΦT d3 x = − VR −
𝜌 𝜕 Φ d3 x
∫
∫ R t T
dt
dt
(38)
(39)
where we made use of the material time derivative, and again used conservation of
mass in the second step. The first way is slightly objectionable since it assumes that
the mass distribution is smooth, whereas in fact we have chosen it to be discontinuous. This can be repaired by providing the bodies with a smooth skin. There is no
such objection to the second way, and we will see that the two ways of calculation
give the same result.
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Pursuing the first way of calculation we note that in the chosen inertial system
the time dependence in 𝜌R enters only through the function QR (t). Using the translation theorem we find that
∫
𝜕t 𝜌R (x)ΦT (x + D)d3 x
(
)
9QT
MD2 − 3QR Q̇ R
GM Q̇ R
1
−
=
−GM
.
=−
2 D3
2D2
2MD2 − 15QR D3
Ẇ R =
(40)
In the last step we used the constraint (35) between the quadrupole moments.
Similarly
Ẇ T = −GM
MD2 − 3QT Q̇ T
.
2MD2 − 15QT D3
(41)
Using the constraint (36) we find (after some calculation) that
̇
Ẇ R + Ẇ T = F,
(42)
where
F=
9GQR QT GM(QR + QT )
GM 2
−
= VR +
.
5
2
D
4D
2D
(43)
If we integrate to find the total amount of work transmitted during a cycle we find
WR + WT =
∮
dF = 0.
(44)
Hence all of the energy transmitted by Tweedledee is received by Tweedledum.
For the second way of calculation it is convenient to choose an inertial system
in which the transmitter is momentarily at rest. Then we are no longer calculating
the same thing. Denoting the rate of work done on the receiver in an inertial system where the transmitter is momentarily at rest by Ẇ ′ , we find
d
Ẇ R� + VR = − 𝜌R (x)𝜕t ΦT (x + D)d3 x = −Ẇ T .
∫
dt
(45)
Comparing the equations obtained, and recalling expression (33) for VR , we see that
they are fully consistent, once we observe that
(
)
d GM 2
�
̇
̇
.
WR = WR +
(46)
dt
D
The motion of the centre of mass explains the difference.
How much energy is being transmitted? With QR = QR (t) we get
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WR =
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2MD2 − 6QR dQR
GM
.
Ẇ R dt =
∮
2 ∮ 2MD2 − 15QR D3
15
(47)
Tweedledum’s aim is to maximize this expression. We see that it can be made positive by letting the function QR (t) lag behind the periodic function D = D(t) so that its
derivative is negative when D is close to its minimum. and positive when D is close
to its maximum. This will ensure that the total amount of work on the receiver in a
cycle is positive. A natural choice is to let both bodies be spherical at the moment of
closest approach, and at the moment when they are at maximal distance from each
other.
From the solution of the two-body problem we know that
D = D(𝜑(t)) =
Dmin (1 + e)
1 + e cos 𝜑
(48)
where e is the eccentricity of the ellipses and 𝜑 is an angular coordinate with respect
to the centre of mass. For definiteness we let Tweedledum choose
QR = −Qmax
sin 𝜑.
R
(49)
The work integral (47) is easily approximated if the bodies are small compared to
the distance between them, that is if QR << MD2. Then we obtain the positive result
max
dQR
GM
3𝜋 e(4 + e2 ) GMQR
.
WR ≈ −
=
2 ∮ D3
8 (1 + e)3 D3
min
(50)
The function of e that occurs here has a maximum at e = 2∕3.
The full work integral (47) is best treated numerically. It can be seen that it is
always positive and that it diverges as the limit 2D2 M = 15QR is approached [24].
Thus it is clear that the twins can achieve their aim of transmitting energy from the
one to the other through the gravitational field.
4 Energy Transfer
Reading Bondi and McCrea behind a veil of hindsight, and knowing that no-one
did more than Bondi to prove that gravitational waves are real and do carry energy
away, it is easy to read their paper as an argument for the reality of gravitational
waves. This is probably a misreading of history though, since Bondi approached
that problem in the best scientific tradition, where nothing is taking for granted
[28]. In 1957 we find him arguing against any glib analogy to the simpler theory of
electrodynamics:
“The cardinal feature of electromagnetic radiation is that when radiation is produced the radiator loses an amount of energy which is independent of the location of
the absorbers. With gravitational radiation, on the other hand, we still [in 1957] do
not know whether a gravitational radiator transmits energy whether there is a near
receiver or not.” [29]
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Two years later, when Bondi and McCrea wrote their paper, the question of the
reality of gravitational waves was still on their minds. But the electrodynamic analogy of their thought experiment is not to electromagnetic radiation, the analogy is
to inductive energy transport in the near zone such as occurs in a transformer. If the
receiver is not there the sender simply stores some energy in the magnetic field, and
gets it back when the AC current is turned off.
However, our reason for revisiting the Bondi–McCrea example was that we
wanted to see how the various candidates for a local gravitational energy density
fare in a concrete example. Equation (21) makes it clear that each choice of energy
density comes with a gravitational Poynting vector describing energy transport.
Thus, outside matter, the energy density proposed by Lynden-Bell and Katz leads to
the Poynting vector
Si = −
1
𝜕 Φ𝜕 Φ
4𝜋 t i
(51)
The energy densities preferred by Maxwell and by Bondi lead to the respective
Poynting vectors
1
Φ𝜕 𝜕 Φ
4𝜋 i t
(52)
1
(Φ𝜕i 𝜕t Φ − 𝜕t Φ𝜕i Φ).
8𝜋
(53)
S�i =
S��i =
In Sect. 2 we made the choice between them by deciding what boundary conditions
to use in order to control the system. But this is not at all how the concrete problem is posed. The gravitational potential is controlled from inside the system by
Tweedledum, and by his agreement with Tweedledee.
However, among the three, Bondi’s Poynting vector S′′i enjoys the advantage that
it is divergence free in vacuum. This lends a special significance to the flux integral
I=
∮S
S��i dSi ,
(54)
where S is a closed surface in vacuum. The surface can be freely deformed within
the vacuum region without changing the value of the integral. If there is vacuum
outside the surface it evaluates to zero. There is no net flux out of the surface,
regardless of how the body inside the surface is changing its multipole moments.
But if the surface divides two regions containing two distinct bodies it quantifies the
amount of energy transferred between them [2]. This means that Bondi’s Poynting
vector offers the best way of calculating the energy transferred between Tweedledum
and Tweedledee. Nevertheless the necessary calculations are very complex. They
are discussed in Ref. [24].
In conclusion, we have found that there is a sense in which Newtonian gravity
prefers a local energy density that is in agreement with that of Lynden–Bell and
Katz [8], but it is notable that this plays no useful role in the concrete discussion of
energy transfer between Tweedledee and Tweedledum. There it seems much more
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Foundations of Physics
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15
helpful to regard the energy as localized within the matter. In the end, the work done
on the body is independent of the way in which gravitational energy is localised
[30]. In general relativity the question of how to define a useful notion of quasilocal energy has given rise to a large literature [31], while the definition of the total
energy of an isolated system is clear [32]. The warning to relativists is not to expect
a unique answer, rather we expect several different answers that are useful in different ways. The question should be: What quasi-local energy expression is best suited
to describe energy transport in a given concrete situation?
Acknowledgements It is a pleasure to thank the referees for their thoughtful comments.
Funding Open access funding provided by Stockholm University.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
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ses/by/4.0/.
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