arXiv:1811.10602v2 [physics.class-ph] 10 Jan 2019
The Mass of the Gravitational Field
Charles T. Sebens
California Institute of Technology
January 10, 2019
arXiv v.2
Forthcoming in
The British Journal for the Philosophy of Science
Abstract
By mass-energy equivalence, the gravitational field has a re
lativistic mass
density proportional to its energy density. I seek to better
understand this mass of
the gravitational field by asking whether it plays three trad
itional roles of mass: the
role in conservation of mass, the inertial role, and the role
as source for gravitation.
The difficult case of general relativity is compared to the mor
e straightforward cases
of Newtonian gravity and electromagnetism by way of gravito
electromagnetism, an
intermediate theory of gravity that resembles electromagn
etism.
Contents
1 Introduction
2
2 The Mass of the Electromagnetic Field
4
2.1 The Conservational Role . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
5
2.2 The Inertial Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
7
2.3 The Gravitational Role . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
9
3 The Mass of the Gravitational Field in Newtonian Gravity
9
3.1 The Conservational Role . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
10
3.2 The Inertial Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
12
3.3 The Gravitational Role . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
13
4 The Mass of the Gravitational Field in Gravitoelectromagn
etism
14
4.1 The Conservational Role . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
16
4.2 The Inertial Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
17
4.3 The Gravitational Role . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
19
5 The Mass of the Gravitational Field in General Relativity
20
5.1 The Conservational Role . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
22
5.2 The Inertial Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
26
5.3 The Gravitational Role . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
27
6 Conclusion
29
A Mass-Energy Equivalence
30
B Deriving Gravitoelectromagnetism from General Relativi
ty
31
1
1 Introduction
By mass-energy equivalence (
E
=
mc
2
), the gravitational field has a relativistic mass
density proportional to its energy density. I seek to better unde
rstand this mass of
the gravitational field by asking whether it plays the traditional role
s of mass—asking
whether the gravitational field really acts like it has mass.
This paper is organized into sections focusing on four different phys
ical theories:
electromagnetism, Newtonian gravity, gravitoelectromagnetism,
and general relativity.
For each theory, I ask whether the field has a mass playing any or all
of the following
three roles: the conservational role (in ensuring conservation of
mass), the inertial role
(in quantifying resistance to acceleration), and the gravitational
role
1
(as source of
gravitation). Here I summarize the results.
The electromagnetic field possesses a mass playing all three of the a
bove roles and
thus serves as a useful point of comparison for analyzing the grav
itational field. The
mass of the electromagnetic field plays the conservational role in en
suring that—although
charged matter will generally gain and lose (relativistic) mass in intera
cting with the
electromagnetic field—the total mass of field and matter is always co
nserved. The fact
that the mass of the electromagnetic field plays the inertial role is of
ten illustrated in an
indirect way: a charged body requires more force to accelerate th
an an uncharged body
because there is additional mass in the electromagnetic field surrou
nding the charged
body. However, this effect is complicated by the fact that accelera
ted charged bodies
emit electromagnetic radiation. We can see the inertial role more dire
ctly by giving a
force law that describes the reaction of the electromagnetic field t
o the forces exerted
upon it by matter. In general relativity, the mass of the electroma
gnetic field clearly
plays the gravitational role.
In Newtonian gravity, the gravitational field does not possess a ma
ss playing any of
the above three roles. One does not need to attribute any mass to
the gravitational field
to ensure conservation of mass. However, one does need to attr
ibute negative energy to
the gravitational field in order to ensure conservation of energy.
There are a number of
ways to do so, one of which
Maxwell
(
[1864]
) found by analogy with electromagnetism.
The Newtonian gravitational field carries no momentum and thus has
no mass playing
the inertial role. The Newtonian gravitational field does not act as a
source for itself
and thus has no mass playing the gravitational role.
Gravitoelectromagnetism is a theory of gravity that can be arrived
at as an extension
of Newtonian gravity (as was first done by
Heaviside
,
[1893]
) or as a limit of general
relativity (as is done in some textbooks). Gravitoelectromagnetism
gets its name from
the fact that the laws of the theory are structurally a very close m
atch to the laws of
1
This role of mass as source of gravitation is sometimes calle
d the ‘active’ gravitational role to
distinguish it from the ‘passive’ gravitational role of mas
s in quantifying the amount of force felt from a
given gravitational field. To limit the scope of this article
, the passive gravitational role is not examined
here.
2
electromagnetism. Because of this close match, it is straightforwa
rd to show that in this
theory the gravitational field possesses a mass playing the conser
vational and inertial
roles. The inertial role played by this mass can be seen directly in the f
orce law for the
field and indirectly in the fact that massive bodies are surrounded by
clouds of negative
field mass which make them easier to accelerate. The force required
to accelerate a body
is also modified by the presence of gravitational radiation which carr
ies away negative
energy. In gravitoelectromagnetism—as it is standardly presente
d—the gravitational
field does not act as a source for itself and thus does not play the gr
avitational role.
However, one can modify the theory (making it nonlinear) so that th
e field is a source
for itself. The gravitational field around a planet is thus slightly weak
ened as it is
now sourced by both the positive mass of the planet itself and the ne
gative mass of its
surrounding gravitational field. Gravitoelectromagnetism is not ne
arly as well-known or
well-studied as the other theories discussed in this paper, but I’ve in
cluded it as it serves
as a useful bridge linking electromagnetism, Newtonian gravity, and
general relativity.
In general relativity there are multiple mathematical objects that
can be used
to describe the flow of energy and momentum (related to the differe
nt energy
densities available in Newtonian gravity), including the Weinberg and La
ndau-Lifshitz
energy-momentum tensors. I discuss these two tensors and call
for another which better
aligns with the three other physical theories discussed above. How
ever one describes the
flow of energy, the mass of the gravitational field plays the conser
vational role. It also
plays the inertial role in a similar way to the mass of fields in the other th
eories. The
mass of the gravitational field appears to play the gravitational ro
le, though it is difficult
to say how it does so as this seems to depend on the way Einstein’s field
equations are
written. In my treatment of general relativity I adopt a field-theo
retic approach to the
theory (as is done in the textbooks of
Weinberg
,
[1972]
and
Feynman
et al.
,
[1995]
) in
order to stress the connections between general relativity and t
he three other theories.
The questions about the mass of the gravitational field pursued in t
his paper are
relevant to a number of ongoing debates in the foundations of phys
ics. First, there
has been much discussion about how to properly understand mass-
energy equivalence,
as expressed mathematically by Einstein’s famous
E
=
mc
2
.
2
Material bodies
clearly possess mass and—by mass-energy equivalence—also posse
ss an energy that
is proportional to their mass. Fields clearly possess energy and—by
mass-energy
equivalence—also possess a mass that is proportional to their ener
gy. But, do they really
act like they possess such a mass? That is the core question of this p
aper. Second, there
is an ongoing debate as to the ontological status of fields like the elec
tromagnetic field
and the gravitational field. Are fields real and if so what kind of thing
are they? One
argument that can be given for the reality of a particular field is that
conservation of
energy only holds if one takes the field to be a real thing that posses
ses energy.
3
But,
2
See the references in appendix
A
.
3
See, e.g., (
Lange
,
[2002]
, ch. 5;
Frisch
,
[2005]
, p. 31;
Lazarovici
,
[2017]
, sec. 4.2).
3
some are not convinced.
Lange
(
[2002]
) contends that it is really the possession of proper
mass by a field (not energy) which grounds the best argument for t
he electromagnetic
field’s reality. I will not explicitly address the question of whether field
s are real in the
paper, but I think that everything I do to show that the mass poss
essed by fields acts
just like the mass possessed by ordinary matter suggests that fie
lds are just as real as
matter. Taking the gravitational field to be real in general relativit
y, one might wonder
whether it is a field on spacetime or a part of spacetime.
4
The former perspective will
prove useful for our purposes here. Third, philosophers of phys
ics have recently put
forward functionalist accounts of a number of important concept
s, including probability,
spacetime, and gravitational energy-momentum.
5
According to a functionalist account
of mass, a quantity would count as a mass provided it played certain r
oles like the three
analyzed here. Although I think it is of interest to determine whethe
r these roles are
played independent of any functionalist ambition, one could certainly
build on the work
done in this paper to develop a functionalist account of the mass of fi
elds. Fourth, there
has been much debate as to how to properly understand the energ
y and momentum of
the gravitational field in general relativity.
6
Asking about the mass of the gravitational
field gives a slightly different angle on this well-studied problem (as the fi
eld’s mass
density is proportional to its energy density). In section
5
I present the progress I’ve
made. However, the discussion here is restricted to the field-theo
retic formulation of
general relativity and the bearing of my conclusions on the theory’s
standard geometric
formulation is left unsettled.
This paper is part of a larger research project on the mass of fields
, including recent
papers on the electromagnetic field and the Dirac field (
Sebens
,
[2018]
,
[unpublished]
).
2 The Mass of the Electromagnetic Field
In the context of special relativity, each material body has a veloc
ity-dependent
relativistic mass (proportional to its energy). That mass plays a nu
mber of different
roles. First, it plays the conservational role. Globally, the sum of all
relativistic mass
never changes. Locally, any change in a body’s relativistic mass can b
e attributed to
a (local) exchange of relativistic mass between that body and somet
hing else. Second,
relativistic mass plays the inertial role. This mass quantifies the body
’s resistance to
being accelerated (though the exact sense in which it does so is more
complex than in
pre-relativistic physics, as
~
F
is no longer equal to
m~a
). Third, relativistic mass plays
the gravitational role. In general relativity, it is this mass which act
s as a source for
gravitation. In relativistic contexts I will use ‘mass’ as shorthand f
or ‘relativistic mass’,
4
See the references in footnotes
33
,
35
, and
36
.
5
See (
Wallace
,
[2012]
, ch. 4;
Knox
,
[forthcoming]
;
Baker
,
[unpublished]
;
Read
,
[forthcoming]
).
6
See the references in footnote
39
.
4
as it is relativistic mass and not proper mass which most directly plays t
hese three roles.
7
According to mass-energy equivalence, if something has energy
E
it has mass
E
/c
2
.
The electromagnetic field thus possesses a mass proportional to it
s energy. Upon first
encountering the idea that the electromagnetic field has mass, one
might think that this
must be a very different sort of mass than the mass of an ordinary m
aterial body. It
is not. The mass of the electromagnetic field plays all three of the ab
ove roles. In this
section I explain how it does so. (See appendix
A
for more on mass-energy equivalence
and
Sebens
,
[2018]
for more on the inertial role played by the electromagnetic field’s
mass.)
Before we start on all of that, let me pause to preempt a potential
confusion.
When particle physicists discuss the ‘mass’ of a given field, they are u
sually talking
about a certain quantity which appears in the dynamical equations f
or the field and
corresponds to the proper mass of the particle associated with th
at field.
8
In this
sense, the electromagnetic field is massless because the photon ha
s no proper mass (in
contrast to, for example, the Dirac field). That is not the sense of
field ‘mass’ which I am
examining here. When I talk about the mass of a field I am talking about
the relativistic
mass
9
of the field, proportional to the field’s energy. Even though the ph
oton has no
proper mass, the electromagnetic field still has a relativistic mass de
nsity equal to its
energy density divided by
c
2
.
2.1 The Conservational Role
We must attribute energy to the electromagnetic field to ensure co
nservation of energy
in electromagnetism—the energy of charged matter alone is not con
served. As mass is
proportional to energy, the mass of matter alone is similarly not con
served. However,
if we attribute mass to the electromagnetic field in proportion to its e
nergy, the total
mass of matter and field is conserved.
The laws of electromagnetism are Maxwell’s equations (
27
) and the Lorentz force
law (
28
). From these laws, one can derive an equation for the conservatio
n of energy
(Poynting’s theorem),
∂
∂t
1
8
π
E
2
+
B
2
+
~
∇ ·
~
S
f
=
−
~
f
f
·
~v
q
m
.
(1)
The first term gives the rate at which the energy of the electromag
netic field,
ρ
E
f
=
1
8
π
E
2
+
B
2
,
(2)
7
Although some other authors use ‘mass’ as shorthand for rela
tivistic mass (e.g.,
Bondi and Spurgin
,
[1987]
), many think that proper mass is more deserving of the title (
see
Okun
,
[1989]
;
Taylor and Wheeler
,
[1992]
, pp. 250–251;
Lange
,
[2001]
).
8
It is this sense of field ‘mass’ that is being used when authors
describe the gravitational field of
general relativity as a massless spin-two tensor field (see t
he references in footnote
39
).
9
One might wonder whether there is a proper mass of the field dis
tinct from the proper mass of the
particle associated with the field. This is discussed in (
Lange
,
[2002]
, ch. 8;
Sebens
,
[2018]
, sec. 7).
5
is changing. (The
E
superscript on
ρ
E
f
indicates that this is the density of energy and
the
f
subscript indicates that it is a property of the electromagnetic field
.) The second
term in (
1
) describes the rate at which field energy flows out of a volume in term
s of the
Poynting vector (the energy flux density),
~
S
f
=
c
4
π
~
E
×
~
B .
(3)
The righthand side of (
1
) gives the rate at which energy is transferred from matter to
field (per unit volume). This rate is expressed in terms of the work do
ne by the Lorentz
force density
~
f
f
as
−
~
f
f
·
~v
q
m
=
−
~
J
m
·
~
E
, where
~
J
m
=
ρ
q
m
~v
q
m
is the current density,
ρ
q
m
is the charge density, and
~v
q
m
is the velocity field describing the flow of charge. (The
q
superscript indicates that these quantities describe the flow of ch
arge as opposed to the
flow of mass or energy and the
m
subscript indicates that these are properties of matter.)
Note that attributing energy to the electromagnetic field, as in (
2
), obviates the need to
attribute potential energy to matter which, for example, increas
es as oppositely charged
bodies are pulled away from one another.
10
By mass-energy equivalence (
E
=
mc
2
), the electromagnetic field has a mass density
equal to its energy density (
2
) divided by
c
2
,
ρ
f
=
1
8
πc
2
E
2
+
B
2
.
(4)
We can divide the above energy conservation equation (
1
) everywhere by
c
2
to arrive at
an equation for the conservation of mass,
∂ρ
f
∂t
+
~
∇ ·
~
G
f
=
−
~
f
f
·
~v
q
m
c
2
,
(5)
where
~
G
f
is the momentum density of the electromagnetic field, equal to the e
nergy flux
density divided by
c
2
:
~
G
f
=
~
S
f
c
2
=
1
4
πc
~
E
×
~
B .
(6)
Thinking of momentum as relativistic mass times velocity, we can write
~
G
f
as
ρ
f
~v
f
by
introducing a velocity
~v
f
to describe the flow of the electromagnetic field’s mass,
11
~v
f
=
~
G
f
ρ
f
=
~
S
f
ρ
E
f
= 2
c
~
E
×
~
B
(
E
2
+
B
2
)
.
(7)
10
Lange
(
[2002]
, ch. 5) discusses why the non-instantaneous nature of elect
romagnetic interactions
makes the inclusion potential energy insufficient to achieve
conservation of energy in electromagnetism
(i.e., why one must attribute an energy density to the electr
omagnetic field as in (
2
)). On p. 119 he
mentions that such an introduction of field energy (replacin
g potential energy) is not necessitated in
Newtonian gravitation, though in section
3
I will adopt such a picture as it better aligns with the other
theories considered here.
11
See (
Poincar ́e
,
[1900]
;
Kraus
,
[1953]
, p. 373;
Geppert
,
[1965]
;
Arora and Geppert
,
[1967]
;
Born and Wolf
,
[1970]
, sec. 14.2.1;
Misner
et al.
,
[1973]
, p. 122;
Sebens
,
[2018]
).
6
This velocity cannot exceed the speed of light (as it attains its maximu
m value,
c
, when
~
E
and
~
B
are perpendicular and equal in magnitude).
2.2 The Inertial Role
The inertial role of mass is its role in quantifying the amount of acceler
ation a body
experiences in response to a given force. The fact that the electr
omagnetic field possesses
such a mass can be seen in an indirect but compelling way by observing t
hat it is harder
to accelerate a charged body (which carries along some field mass wit
h it) than an
otherwise similar uncharged body (which carries none). The appare
nt inertial mass of
the charged body is larger because both the mass of the field and th
e mass of matter
are resisting acceleration. But, this is only one of two effects modify
ing the way a
charged body reacts to forces (as compared to an uncharged bo
dy). Charged bodies
also experience radiation reaction forces when they are accelerat
ed because they emit
electromagnetic radiation which carries away energy and momentum
.
12
The radiation
reaction force may point opposite the acceleration (making the bod
y even harder to
accelerate) or it may point in some other direction. The total force
on the body resulting
from these two distinct effects can be called the ‘field reaction’.
13
For an ordinary
macroscopic charged object, the field reaction is insignificant beca
use the amount of
energy in the electromagnetic field is negligible. However, for subato
mic particles (like
the electron) the field reaction becomes quite important.
As a simple example of field reaction, consider a spherical positively ch
arged body
which is acted upon by an applied force that uniformly accelerates it f
rom rest over a
very short period of time and then ceases. The electric field around
such a body—after
the acceleration has finished but before the radiation has escaped
too far—is depicted
in figure
1
. Before and after the acceleration, the charge is surrounded by
an outwardly
directed electric field (and after by a magnetic field as well). Interpo
lating between
these pre- and post-acceleration electric fields is a quite differently
oriented electric field
resulting from the period of acceleration. During the period of acce
leration, a field like
this passes through the body and—as can be seen by looking at the d
irection of the field
lines in the figure—the net electric force points opposite the directio
n of acceleration,
making it more difficult to accelerate the charge. Both of the effects
mentioned in the
previous paragraph contribute to making the charge more difficult t
o accelerate: The
12
For an introduction to radiation reaction, see (
Pearle
,
[1982]
;
Griffiths
,
[1999]
, ch. 11;
Jackson
,
[1999]
, ch. 16).
13
This terminology follows (
Griffiths
,
[1999]
, sec. 11.2). Griffiths divides the electromagnetic field
surrounding an accelerated charge into a ‘velocity field’ wh
ich stays with the charge as it moves and
an ‘acceleration field’ which radiates off to infinity. He goes
on to use this division of the field into two
parts to explain the difference between field reaction and rad
iation reaction: ‘As the particle accelerates
and decelerates energy is exchanged between it and the veloc
ity fields, at the same time as energy is
irretrievably radiated away by the acceleration fields. ...
if we want to know the recoil force exerted by
the fields on the charge, we need to consider the
total
power lost at any instant, not just the portion
that eventually escapes in the form of radiation. (The term “
radiation reaction” is a misnomer. We
should really call it
field reaction
. ...)’
7
applied force must accelerate not just the mass of body itself, but
the mass of the field
surrounding it as well. Also, the applied force must provide the energ
y which is radiated
away in electromagnetic waves.
Figure 1: This figure shows the electric field lines around a positive cha
rge that was
initially at rest and then for a brief period quickly accelerated to half t
he speed of light.
The dashed circle indicates the particle’s initial position. Figures like th
is are discussed
in (
Purcell and Morin
,
[2013]
, sec. 5.7).
The above approach to understanding the inertial role played by th
e electromagnetic
field is indirect. The mass of the field should quantify resistance to ac
celeration of the
thing which possesses that mass—the field itself. Using the field veloc
ity given above
(
7
), it can be shown that the field’s mass does play this role. Consider th
e conservation
of momentum equation for electromagnetism,
−
~
f
f
=
∂
∂t
(
ρ
f
~v
f
)
−
~
∇ ·
↔
σ
f
,
(8)
where
−
~
f
f
is the force density exerted by matter upon the field (equal and op
posite the
force exerted by the field upon matter) and
↔
σ
f
is the momentum flux density tensor for
the electromagnetic field (also known as the Maxwell stress tensor
),
↔
σ
f
=
1
4
π
~
E
⊗
~
E
+
1
4
π
~
B
⊗
~
B
−
1
8
π
E
2
+
B
2
↔
I .
(9)
Upon integrating (
8
) over a volume, the left side gives the force exerted on the field in
that volume, the first term on the right gives the rate at which the m
omentum of the
field in that volume is changing, and the second term on the right gives
the rate at which
field momentum is leaving that volume.
It is not standard to speak, as I just did, of forces acting upon th
e electromagnetic
field. But, I think it is helpful to do so. The response of the electrom
agnetic field to
what I have described as a force exerted by matter has the same f
orm as the relativistic
8
Eulerian force law giving the response of matter (modeled as a contin
uum) to the equal
and opposite force exerted by the field upon matter,
~
f
f
=
∂
∂t
(
ρ
m
~v
m
)
−
~
∇ ·
↔
σ
m
,
(10)
where
ρ
m
is the relativistic mass density,
~v
m
is the velocity of mass flow, and
↔
σ
m
is the
momentum flux density tensor for matter. Comparing (
8
) and (
10
), we see that (
8
) is
an Eulerian force law for the electromagnetic field. In that equation
the mass of the
electromagnetic field quantifies resistance to acceleration in just t
he same way that the
mass of matter quantifies resistance to acceleration in (
10
). To better understand the
inertial role of the field’s mass, we could also analyze the Lagrangian f
orms of these force
laws (which use the material derivative
D
Dt
), though we will not do so here (see
Sebens
,
[2018]
).
2.3 The Gravitational Role
In general relativity, the mass of the electromagnetic field acts as
a source of gravitation
in just the same way that the mass of matter does. In fact, the ele
ctromagnetic field
is considered to be ‘matter’ in the broad way the term is often used in
the context of
general relativity.
3 The Mass of the Gravitational Field in Newtonian
Gravity
In Newtonian gravity the gravitational field does not possess a mas
s playing any of
the three roles we just went through. This should come as no surpr
ise. Mass-energy
equivalence was never a part of Newton’s theory of gravity. Still, it is
worthwhile to see
exactly how the gravitational field fails to play these roles for futur
e comparison between
Newtonian gravity and more advanced theories of gravity.
Let us consider how the Newtonian gravitational field interacts with
a continuous
distribution of matter. The following equation describes how mass ac
ts as a source for
the gravitational field:
~
∇ ·
~g
=
−∇
2
φ
=
−
4
πGρ
m
.
(11)
Here
φ
is the gravitational potential and
~g
is the gravitational field, related to
φ
by
~g
=
−
~
∇
φ
. The density of force exerted upon matter by the gravitational fi
eld is
~
f
g
=
ρ
m
~g
=
−
ρ
m
~
∇
φ
(12)
The
g
subscript indicates that a quantity pertains to the gravitational fi
eld (for example,
~
f
g
is the force exerted by the gravitational field). As we will be conside
ring three distinct
theories of gravitation, be aware that expressions for such quan
tities will change.
9
The mass density
ρ
m
that appears in (
11
) and (
12
) is, of course, the density of
ordinary non-relativistic mass. In the context of Newtonian gravit
y, we will use the
term ‘mass’ to refer to this kind of mass. In the next section, we will
shift back to using
‘mass’ for relativistic mass.
3.1 The Conservational Role
In Newtonian gravity, there is no need to attribute mass to the gra
vitational field in
order to ensure conservation of mass. The (non-relativistic) mas
s of matter is itself
conserved
∂ρ
m
∂t
=
−
~
∇ ·
(
ρ
m
~v
m
)
,
(13)
in contrast to the (relativistic) mass of matter in electromagnetism
—which is only
conserved in conjunction with the (relativistic) mass of the electro
magnetic field, see
(
5
). The gravitational field does not have a mass playing the conserva
tional role.
However, the gravitational field can be attributed an energy dens
ity to ensure
conservation of energy. From (
11
), (
12
), and (
13
), we can derive a conservation of
energy equation for gravity similar to (
1
),
∂
∂t
−
g
2
8
πG
+
~
∇ ·
~
S
g
=
−
~
f
g
·
~v
m
,
(14)
where
ρ
E
g
=
−
g
2
8
πG
=
−|
~
∇
φ
|
2
8
πG
(15)
is interpreted as the energy density of the gravitational field and
~
S
g
as the Poynting
vector for the gravitational field (giving the gravitational energy
flux density),
14
~
S
g
=
1
4
πG
φ
∂
∂t
~
∇
φ
+
~v
m
φ
∇
2
φ
.
(16)
In words, (
14
) says that the rate at which the energy of the gravitational field in
a
volume changes plus the rate at which gravitational energy leaves t
hat volume is equal
to the rate at which energy is transferred from matter to the gra
vitational field within
that volume.
The energy density of the Newtonian gravitational field in (
15
) is quite similar
in form to the energy density of the electromagnetic field (
2
). However, because of
the difference in sign the gravitational energy density is always nega
tive. Noting the
similarities between gravity and electromagnetism,
Maxwell
(
[1864]
, part IV) tentatively
proposed the above expression for the energy density of the gra
vitational field. Maxwell
was troubled by the fact that this energy density is negative. He th
ought that because
‘energy is essentially positive’, it would be ‘impossible for any part of spa
ce to have
14
This Poynting vector appears in (
Synge
,
[1972]
, eq. 5.11, assuming
ρ
m
= 0;
Noonan
,
[1984]
, eq. 10).
10
negative intrinsic energy’ and thus that space must possess an ‘en
ormous [positive]
intrinsic energy’ which (
15
) describes deficits of. We need not share this particular
concern. Still, the idea that gravitational energy may be negative r
aises questions. This
feature of gravitational energy will be of special interest to us in t
he next two theories
of gravity where, by mass-energy equivalence, we will be dealing with
negative mass.
The Poynting vector for Newtonian gravity (
16
) looks odd in comparison to the
Poynting vector for electromagnetism (
3
). The first oddity to note is that
~
S
g
is not
expressed purely in terms of field variables, the velocity of matter
~v
m
appears—as might
ρ
m
if (
16
) were rewritten using (
11
). The second oddity is that
~
S
g
is written in terms of
the gravitational potential
φ
and cannot be rewritten solely in terms of the gravitational
field
~g
. This should raise red flags that the Poynting vector may be gauge-
dependent.
Indeed it is. Should you shift
φ
everywhere by a constant
s
,
~g
will be unchanged
but
~
S
g
will acquire an additional contribution of
s
4
πG
∂
∂t
~
∇
φ
+
~v
m
∇
2
φ
. However, this
additional contribution is divergenceless—as can easily be seen using
(
11
) and (
13
).
Thus, the additional contribution does not change the flux of ener
gy out of any closed
surface and does not appear in (
14
).
15
It is worth noting that although we will continue to use the expressio
ns for energy
density and energy flux in (
15
) and (
16
), they are not unique. Consider the alternative
energy density of
ρ
E
g
=
1
2
ρ
m
φ
=
φ
∇
2
φ
8
πG
,
(17)
which, unlike (
15
), is a gauge-dependent function of the gravitational potential n
ot
expressible in terms of the gravitational field alone. This density can
be interpreted in a
number of ways: as an energy density of the field, an interaction en
ergy density of matter
and field, or a potential energy density of matter. I will treat it as a
n energy density
of the field, like (
2
) and (
15
). One can rewrite the conservation of energy equation (
14
)
using this new energy density and a suitably altered Poynting vector
,
16
∂
∂t
1
2
ρ
m
φ
+
~
∇ ·
1
4
πG
1
2
φ
∂
∂t
~
∇
φ
−
1
2
∂φ
∂t
~
∇
φ
+
~v
m
φ
∇
2
φ
=
−
~
f
g
·
~v
m
.
(18)
As will become relevant in section
5
, it is also possible to form energy densities by
combining contributions from (
15
) and (
17
).
17
The general form of such an energy
density is
ρ
E
g
=
α
−
g
2
8
πG
+ (1
−
α
)
1
2
ρ
m
φ
=
α
−|
~
∇
φ
|
2
8
πG
+ (1
−
α
)
φ
∇
2
φ
8
πG
,
(19)
where
α
can be varied. The conservation of energy equation for such an en
ergy density
15
The possibility of adding divergenceless terms to the elect
romagnetic Poynting vector is discussed
in (
Lange
,
[2001]
;
Lange
,
[2002]
, ch. 5;
Griffiths
,
[1999]
, p. 347, footnote 1;
Jackson
,
[1999]
, sec. 6.7 and
12.10).
16
This Poynting vector appears in (
Bondi
,
[1962]
)—restricted to empty space where
ρ
m
= 0.
17
Such alternative expressions for gravitational energy den
sity are discussed in (
Peters
,
[1981]
;
Ohanian and Ruffini
,
[2013]
, sec. 1.3;
Thorne and Blandford
,
[2017]
, box 13.4).
11
is
∂
∂t
α
−
g
2
8
πG
+ (1
−
α
)
1
2
ρ
m
φ
+
~
∇ ·
1
4
πG
1 +
α
2
φ
∂
∂t
~
∇
φ
+
α
−
1
2
∂φ
∂t
~
∇
φ
+
~v
m
φ
∇
2
φ
=
−
~
f
g
·
~v
m
.
(20)
3.2 The Inertial Role
Just as there is no need to attribute mass to the gravitational field
in order to ensure
conservation of mass, there is also no need to attribute momentum
to the gravitational
field in order to ensure conservation of momentum. The integral of
the force density
(
12
) over all of space is zero and thus the momentum of matter is conse
rved globally.
Local conservation of momentum can be expressed by an equation
much like (
8
),
−
~
f
g
=
−
~
∇ ·
↔
σ
g
,
(21)
where
↔
σ
g
is the momentum flux density tensor for the gravitational field,
18
↔
σ
g
=
−
1
4
πG
~g
⊗
~g
+
1
8
πG
g
2
↔
I .
(22)
Interpreting (
21
) analogously to (
8
), the equation states that any change in the
momentum of matter in a volume is balanced by a flow of momentum into o
r out of
that volume (as captured by the righthand side). Because this bala
nce is exact, there is
never any accumulation of momentum in the gravitational field and th
us the field has
no momentum density (though it does have a momentum flux density s
ince momentum
flows through it). This is in contrast to the electromagnetic field whe
re
~
f
f
does not
exactly balance
~
∇ ·
↔
σ
f
and momentum can be transferred to or from the field (not
merely through it). The fact that momentum does not accumulate in
the gravitational
field can be seen as a result of the instantaneous nature of gravita
tional interactions
between bodies. Because the gravitational field has no momentum,
it does not have any
mass playing the inertial role.
Newtonian gravity is usually understood as a theory in which material
bodies directly
exert forces upon one another. But, the above comparison with e
lectromagnetism
suggests a different picture. Material bodies do not exert forces
directly upon one
another but instead these forces are mediated (though not delay
ed) by the gravitational
field. The gravitational field exerts forces on matter—given by (
12
)—and matter exerts
equal and opposite forces upon the gravitational field—the leftha
nd side of (
21
). On the
righthand side of (
21
),
↔
σ
g
acts as a stress tensor describing the forces exerted within the
field (the field-on-field forces).
19
The fact that the left and right sides of (
21
) are equal
18
This tensor appears in (
Chandrasekhar
,
[1969]
, eq. 11;
Synge
,
[1972]
, eq. 3.2;
Misner
et al.
,
[1973]
, eq. 39.18;
Noonan
,
[1984]
, eq. 4;
Giulini
,
[1997]
, eq. 2.4;
Straumann
,
[2004]
, eq. 5.61;
Thorne and Blandford
,
[2017]
, box 13.4).
19
In (
Sebens
,
[2018]
) I distinguished between the momentum flux density for the el
ectromagnetic field
12
means that the net force on the field in any region of space is zero. T
hus, the field acts
somewhat like an idealized massless string, which transmits forces bu
t does not acquire
momentum because the forces that act upon any bit of string are n
ever unbalanced.
An example may help to illustrate the above understanding of forces
in Newtonian
gravity. Suppose there are two massive bodies separated from ea
ch other by some
distance. Each will experience a force from the gravitational field d
irected towards the
other body and each will exert a force on the gravitational field dire
cted away from the
other body. These forces on the gravitational field will be balanced
by tensile forces in
the gravitational field connecting the forces exerted by the two b
odies.
20
This is similar
to what happens in the electrostatic case of two opposite charges
held in place some
distance from one another. Each experiences an electric force fr
om the electromagnetic
field directed towards the other body and each exerts a force on t
he electromagnetic field
directed away from the other body. Tensile forces in the electroma
gnetic field connect
the forces exerted by the two bodies upon the field. In this scenar
io, the electromagnetic
field carries no momentum—as can be seen immediately from (
6
), noting that there is
no magnetic field.
When discussing the inertial role of field mass in electromagnetism we b
egan by
addressing the problem indirectly, asking whether it is more difficult to
accelerate a
charged body from rest than an uncharged body. Similarly, one cou
ld ask if it is any
more difficult to accelerate a body along with the gravitational field su
rrounding it than
it would be to accelerate the body without its gravitational field (imag
ining gravity to
be turned off). Of course, it is no more difficult. The gravitational fie
ld around a body
does not contribute to its apparent inertial mass in Newton’s theor
y of gravity.
3.3 The Gravitational Role
In (
11
), the only source for gravity is the mass of matter,
ρ
m
. One could imagine
modifying this equation so that gravity acts as a source for itself, a
dding to the righthand
side the energy density of the gravitational field divided by
c
2
.
21
In light of mass-energy
equivalence, this seems like a natural move. But, mass-energy equ
ivalence is not part
of Newtonian gravity and so we will delay consideration of such a move
until the next
section where we examine a theory of gravity that includes mass-en
ergy equivalence. For
now, looking at Newtonian gravity as it is, the gravitational field has n
o mass playing
the gravitational role.
and the true stress tensor (which is not the Maxwell stress te
nsor). We need not make such a distinction
in the context of Newtonian gravity since the
ρ
g
(
~v
g
⊗
~v
g
) term that would differentiate the two tensors
is zero (because the gravitational field has no mass).
20
Such ‘gravitational tensions’ are mentioned by
Kaplan
et al.
(
[2009]
, sec. 4) in the context of
gravitoelectromagnetism. The idea of tensions in the elect
romagnetic field is common, dating back
to Faraday’s work (see, e.g.,
Whittaker
,
[1951]
, pp. 187, 271–273;
Misner
et al.
,
[1973]
, sec. 5.6;
Sebens
,
[2018]
, sec. 5).
21
A number of authors have considered such a variant of Newtoni
an gravitation, including
Einstein
(
[1912]
);
Geroch
(
[1978]
);
Peters
(
[1981]
);
Visser
(
[1989]
);
Giulini
(
[1997]
);
Jefimenko
(
[2000]
,
[2006]
);
Franklin
(
[2015]
).
13
4 The Mass of the Gravitational Field in
Gravitoelectromagnetism
Gravitoelectromagnetism is an intermediate theory of gravity that
serves as a useful
stepping stone in moving from Newtonian gravity to general relativit
y and is thus
sometimes included in textbooks on general relativity. Gravitoelect
romagnetism
is not nearly as empirically successful as general relativity, but it co
ntains some
significant improvements over Newtonian gravitation: the theory a
ccurately predicts
Lense-Thirring precession and introduces a finite speed at which gr
avity propagates
(because the theory includes gravitational waves which act like elec
tromagnetic waves).
Studying gravitoelectromagnetism allows us to bring together what
we’ve learned about
mass in the electromagnetic and gravitational fields on our way to un
derstanding the
mass of the gravitational field in general relativity. Gravitoelectro
magnetism can be
arrived at either as an extension of Newtonian gravity modeled off ele
ctromagnetism or
as a weak-field slow-velocity approximation to general relativity.
22
I present the first
method here and the second in appendix
B
.
If the relevant velocities are sufficiently small, the laws of electromag
netism, (
27
) and
(
28
), reduce to the laws of electrostatics,
Electrostatics
~
∇ ·
~
E
= 4
πρ
q
m
~
∇ ×
~
E
= 0
(23)
~
f
f
=
ρ
q
m
~
E ,
(24)
because there is no appreciable magnetic field produced by moving ch
arges.
23
These
22
For more on gravitoelectromagnetism as an extension of Newt
onian gravity, see (
Heaviside
,
[1893]
;
Misner
et al.
,
[1973]
, ex. 7.2;
Jefimenko
,
[2000]
,
[2006]
). For more on gravitoelectromagnetism as a limit
of general relativity and on the empirical observation of gr
avitomagnetic effects, see (
Forward
,
[1961]
;
Braginsky
et al.
,
[1977]
;
Wald
,
[1984]
, sec. 4.4a;
Harris
,
[1991]
;
Damour
et al.
,
[1991]
;
Jantzen
et al.
,
[1992]
;
Ciufolini and Wheeler
,
[1995]
;
Maartens and Bassett
,
[1998]
;
Mashhoon
et al.
,
[1999]
,
[2001]
;
Mashhoon
,
[2001]
,
[2007]
;
Clark and Tucker
,
[2000]
;
Straumann
,
[unpublished]
;
Tartaglia and Ruggiero
,
[2003]
;
Hobson
et al.
,
[2006]
, app. 17A;
Kaplan
et al.
,
[2009]
;
Keppel
et al.
,
[2009]
;
Ohanian and Ruffini
,
[2013]
, sec. 3.4 and 4.7;
Bakopoulos and Kanti
,
[2017]
). In this literature there is wide variation
regarding notation and definitions. I see this chaos as a lice
nse to define things as I see fit. I’ve
chosen to define the gravitoelectric and gravitomagnetic fie
lds
~
E
g
and
~
B
g
so that they have the same
units as
~
E
and
~
B
. This choice makes the expressions for energy density and mo
mentum flux density
particularly simple. As compared to (
Jefimenko
,
[2000]
,
[2006]
), my gravitoelectric field is
1
√
G
times his
gravitational field
~g
and my gravitomagnetic field is
c
√
G
times his cogravitational field
~
K
. As compared
to (
Hobson
et al.
,
[2006]
, app. 17A), my gravitoelectric field is
1
√
G
times theirs and my gravitomagnetic
field is
c
4
√
G
times theirs. As compared to (
Mashhoon
et al.
,
[1999]
,
[2001]
;
Mashhoon
,
[2001]
,
[2007]
),
my gravitoelectric field is
−
1
√
G
times theirs and my gravitomagnetic field is
−
1
2
√
G
times theirs.
23
From these laws of electrostatics, one could derive a conser
vation of energy equation similar to (
14
)
(with
φ
replaced by the electric potential
V
). In so doing, a Poynting vector for the electric field would
be introduced with the same defects as (
16
). As was the case for the Newtonian gravitational field, the
energy density of the electric field would not be unique. For e
xample, the energy density
1
2
ρ
q
m
V
, like
(
17
), could be used in place of
E
2
8
π
from (
2
) (
Griffiths
,
[1999]
, eq. 2.43;
Jackson
,
[1999]
, eq. 1.53;
Peters
,
14
laws closely resemble the laws of Newtonian gravity, (
11
) and (
12
) along with
~
∇ ×
~g
= 0
(which follows from the fact that
~g
is the gradient of a scalar field),
Gravitostatics
~
∇ ·
~
E
g
=
−
4
π
√
Gρ
m
~
∇ ×
~
E
g
= 0
(25)
~
f
g
=
√
Gρ
m
~
E
g
.
(26)
Here we’ve replaced
~g
by the rescaled
~
E
g
=
1
√
G
~g
(the ‘gravitoelectric field’) and titled
the theory ‘gravitostatics’ to suggest that it is the low-velocity limit
of a deeper theory.
The difference between gravitostatics and electrostatics is just t
hat the charge density
which generates the field has been replaced by
−
√
G
times the mass density and the
charge density used for calculating the force exerted by the field h
as been replaced by
√
G
times the mass density (these differing signs ensure that gravity is a
ttractive and
not repulsive).
24
We can extend this replacement procedure to full electromagnetis
m
in order to get a theory of gravitoelectromagnetism in which moving m
ass produces a
‘gravitomagnetic field’ just as moving charge produces a magnetic fi
eld. (Of particular
importance for applications of the theory is the fact that rotating
planets produce
gravitomagnetic fields which are measurable, but weak enough that
their effects don’t
spoil the successes of Newtonian gravity.) Here are the Maxwell eq
uations and Lorentz
force laws for the two theories,
25
placed side-by-side to facilitate quick comparison:
Electromagnetism
Gravitoelectromagnetism
~
∇ ·
~
E
= 4
πρ
q
m
~
∇ ·
~
E
g
=
−
4
π
√
Gρ
m
~
∇ ×
~
E
=
−
1
c
∂
~
B
∂t
~
∇ ×
~
E
g
=
−
1
c
∂
~
B
g
∂t
~
∇ ·
~
B
= 0
~
∇ ·
~
B
g
= 0
~
∇ ×
~
B
=
4
π
c
~
J
m
+
1
c
∂
~
E
∂t
~
∇ ×
~
B
g
=
−
4
π
c
√
G
~
G
m
+
1
c
∂
~
E
g
∂t
(27)
~
f
f
=
ρ
q
m
~
E
+
1
c
~v
q
m
×
~
B
~
f
g
=
√
Gρ
m
~
E
g
+
4
c
~v
m
×
~
B
g
.
(28)
Moving further away from Newtonian gravity, I will take the
ρ
m
which appears in
the laws of gravitoelectromagnetism to be the relativistic mass dens
ity of matter and
henceforth resume the use of ‘mass’ as shorthand for ‘relativistic
mass’. The momentum
density of matter is
~
G
m
=
ρ
m
~v
m
. Just as the electric and magnetic fields are referred to
[1981]
).
24
We can thus think of mass as a kind of ‘gravitational charge’.
Or, following Coulomb, we could go
the other way and think of electric charge as a kind of ‘electr
ical mass’ (
Roller and Roller
,
[1954]
, p.
79).
25
Here I treat both fields as interacting only with matter and ig
nore any interactions between the
electromagnetic and gravitational fields.
15
together as the electromagnetic field, the gravitoelectric and gra
vitomagnetic fields will
be referred to together as the gravitational field (which is much les
s of a mouthful than
‘the gravitoelectromagnetic field’).
Note that (
27
) and (
28
) include one important deviation from the recipe of
substitution given above: there is a factor of 4 which accompanies t
he gravitomagnetic
force on matter in
~
f
g
but not the magnetic force on matter in
~
f
f
. One would not
have guessed this factor of 4 from the comparison with electromag
netism (and for that
reason it does not appear in all presentations of gravitoelectroma
gnetism
26
). The factor
of 4 arises when you derive gravitoelectromagnetism as an approxim
ation to general
relativity (appendix
B
) and is important to the theory’s applications (
Straumann
,
[unpublished]
, sec. 2.3). As we will see, this disanalogy with electromagnetism leads
to a violation of momentum conservation. So, at times it will be helpful
to compare
gravitoelectromagnetism (as presented above) with an alternativ
e theory in which the
troublesome factor of 4 is not present.
4.1 The Conservational Role
From the gravitoelectromagnetic Maxwell equations and Lorentz f
orce law, one can
derive an equation for the conservation of energy analogous to Po
ynting’s theorem in
electromagnetism (
1
),
∂
∂t
−
1
8
π
E
2
g
+
B
2
g
+
~
∇ ·
~
S
g
=
−
~
f
g
·
~v
m
.
(29)
In this equation the aforementioned factor of 4 is irrelevant since t
he gravitomagnetic
force drops out when
~
f
g
is dotted with
~v
m
. The energy density of the gravitational field
which appears in (
29
) is
ρ
E
g
=
−
1
8
π
E
2
g
+
B
2
g
,
(30)
and the Poynting vector for the gravitational field is
27
~
S
g
=
−
c
4
π
~
E
g
×
~
B
g
.
(31)
Dividing (
29
) by
c
2
yields an equation for the conservation of mass similar to (
5
),
∂ρ
g
∂t
+
~
∇ ·
~
G
g
=
−
~
f
g
·
~v
m
c
2
,
(32)
where the mass density of the field is
ρ
g
=
−
1
8
πc
2
E
2
g
+
B
2
g
,
(33)
26
See, e.g., (
Jefimenko
,
[2000]
,
[2006]
).
27
This Poynting vector appears in (
Heaviside
,
[1893]
;
Jefimenko
,
[2000]
, eq. 6-2.41;
Mashhoon
,
[2007]
,
eq. 3.17).
16
and
~
G
g
is the momentum density arrived at by diving the Poynting vector by
c
2
,
~
G
g
=
−
1
4
πc
~
E
g
×
~
B
g
.
(34)
We can introduce a velocity for the gravitational field,
~v
g
=
~
G
g
ρ
g
=
~
S
g
ρ
E
g
= 2
c
~
E
g
×
~
B
g
E
2
g
+
B
2
g
,
(35)
which takes exactly the same form as (
7
) and is similarly capped at
c
. Note that the
mass and energy of the gravitational field are negative. This is an imp
ortant disanalogy
with electromagnetism, the consequences of which will be explored s
hortly.
Before proceeding to consider the inertial role of the gravitationa
l field’s mass
in gravitoelectromagnetism, let’s briefly discuss the connection bet
ween the above
equations for gravitoelectromagnetism and the corresponding eq
uations for Newtonian
gravity. The energy density in (
30
) is a straightforward extension of the energy density
in (
15
) to incorporate the new gravitomagnetic field. The Poynting vecto
r in (
31
) looks
quite different from (
16
) and does not have the same defects. Still, the divergence of
~
S
g
—which appears in (
29
) and determines the flux of gravitational energy through any
closed surface—will agree with the Newtonian expression in the appr
opriate limit.
28
4.2 The Inertial Role
As before, one can approach the question of whether the gravita
tional field has a mass
playing the inertial role indirectly by asking how difficult it is to accelerat
e a massive
body along with its gravitational field as compared to accelerating th
e same body
with gravity turned off. In gravitoelectromagnetism, the fact tha
t massive bodies are
accompanied by clouds of negative field mass makes them easier to ac
celerate. However,
as in electromagnetism, there is a second effect to consider: radiat
ion reaction. In
gravitoelectromagnetism, accelerating massive bodies gain energy
from the emission of
negative energy gravitational radiation.
Consider the simple case of applying an external force to accelerat
e a massive
28
Starting with
~
S
g
from (
31
), the divergence is
~
∇ ·
~
S
g
=
−
c
4
π
(
~
B
g
·
(
~
∇ ×
~
E
g
)
−
~
E
g
·
(
~
∇ ×
~
B
g
)
)
.
(36)
In the Newtonian limit the curl of
~
E
g
is zero (
25
). We can use the gravitoelectromagnetic Maxwell
equations (
27
) to expand the curl of
~
B
g
,
~
∇ ·
~
S
g
=
1
4
π
~
E
g
∂
~
E
g
∂t
−
√
Gρ
~
E
g
·
~v
m
.
(37)
Using (
11
) and (
13
), this becomes
~
∇ ·
~
S
g
=
~
∇ ·
[
1
4
πG
φ
∂
∂t
~
∇
φ
+
1
4
πG
~vφ
~
∇
2
φ
]
,
(38)
in agreement with (
16
).
17
spherical body from rest. As in section
2.2
, we can see that there will be a field
reaction force opposing that acceleration by looking at the field lines
. Because Maxwell’s
equations are essentially unchanged, figure
1
will accurately depict the gravitoelectric
field around a body that was temporarily accelerated from rest pro
vided we flip the
arrows on the field lines. The field reaction force will thus point along t
he direction of
acceleration, making the body easier to accelerate than it otherwis
e would be.
It is a problematic feature of the theory that the energy of gravit
ational waves is
negative. A physical system could potentially gain arbitrarily large am
ounts of energy
by sending out waves of negative energy into empty space. This pec
uliar feature of
gravitoelectromagnetism is removed in general relativity where—alt
hough the energy of
the gravitational field can still be negative—the energy carried awa
y by gravitational
waves is positive.
Let us now consider the inertial role directly, asking how hard it is to a
ccelerate the
gravitational field itself. From (
27
) and (
28
) you can calculate an Eulerian force law for
the gravitational field,
−
√
Gρ
m
~
E
g
+
1
c
~v
m
×
~
B
g
=
∂
∂t
(
ρ
g
~v
g
)
−
~
∇ ·
↔
σ
g
,
(39)
where
ρ
g
~v
g
is the momentum density of the gravitational field and
↔
σ
g
is the momentum
flux density tensor
↔
σ
g
=
−
1
4
π
~
E
g
⊗
~
E
g
+
−
1
4
π
~
B
g
⊗
~
B
g
+
1
8
π
E
2
g
+
B
2
g
↔
I ,
(40)
which clearly reduces to the Newtonian expression (
22
) when the gravitomagnetic field
is negligible. The key difference between (
39
) and the conservation of momentum
equation for electromagnetism (
8
) is that the force exerted upon the gravitational field
by matter appearing on the left side of (
39
) is not equal and opposite the force exerted
by the gravitational field upon matter (
28
). Because of this violation of Newton’s third
law, momentum is not conserved in gravitoelectromagnetism (as the
theory has been
formulated above). However, if the factor of 4 were not included in
(
28
) (a possibility
discussed earlier) then the forces would balance and momentum wou
ld be conserved in
the theory.
Considering the theory without the factor of 4,
Jefimenko
(
[2006]
, ch. 8) argues that
Newton’s third law does not hold in gravitoelectromagnetism (though
conservation of
momentum does). An apparent violation of Newton’s third law: when t
here are just
two massive bodies in motion, the forces each one feels may not be eq
ual and opposite.
If forces are understood to be exerted by each body on the othe
r, this would amount
to a violation of Newton’s third law. But, if we instead think of forces a
s exerted by
the gravitational field upon matter and by matter back upon the gr
avitational field,
then from (
28
) (without the 4) and (
39
) it is clear that Newton’s third law is upheld.
The same maneuver can be used to save Newton’s third law from a simila
r challenge in
18
electromagnetism (
Sebens
,
[2018]
).
4.3 The Gravitational Role
There is an inconsistency in gravitoelectromagnetism. As the gravit
ational field exerts
forces upon matter, energy will be exchanged between field and ma
tter at a rate of
~
f
g
·
~v
m
(
29
). By mass-energy equivalence, this means that the total mass of
matter
will be changing over time. However, by evaluating
~
∇ ·
(
~
∇ ×
~
B
g
) = 0 using the
gravitoelectromagnetic Maxwell equations (
27
) you can derive an equation requiring
conservation of mass for matter alone—just as one can derive the
conservation of
charge from the electromagnetic Maxwell equations. One way to rid
the theory of
this inconsistency is to say that the mass density which appears in th
e theory’s laws,
(
27
) and (
28
), is the density of proper mass, not relativistic mass. Since proper
mass
will not change as matter gains and loses energy in gravitoelectroma
gnetic interactions,
the proper mass of matter alone would be conserved. However, th
is response takes
gravitoelectromagnetism farther from general relativity, where
it is relativistic mass
(not proper mass) that acts as the source of gravitation. Anoth
er way to cure this
inconsistency is to modify the gravitoelectromagnetic Maxwell equa
tions so that the
mass of the gravitational field itself acts as a source for gravity (t
hereby having the
mass of the gravitational field play all three of its rightful roles, no
t just two):
29
Gravitoelectromagnetism (with self-source terms)
~
∇ ·
~
E
g
=
−
4
π
√
G
(
ρ
m
+
ρ
g
)
~
∇ ×
~
E
g
=
−
1
c
∂
~
B
g
∂t
~
∇ ·
~
B
g
= 0
~
∇ ×
~
B
g
=
−
4
π
c
√
G
~
G
m
+
~
G
g
+
1
c
∂
~
E
g
∂t
(41)
~
f
g
=
√
Gρ
m
~
E
g
+
4
c
~v
m
×
~
B
g
.
(42)
The aforementioned inconsistency is removed as the equation for c
onservation of energy
(
29
) is unchanged and
~
∇ ·
(
~
∇ ×
~
B
g
) = 0 now yields a conservation law for the total mass
of matter and field,
∂ρ
m
∂t
+
∂ρ
g
∂t
=
−
~
∇ ·
(
ρ
m
~v
m
)
−
~
∇ ·
(
ρ
g
~v
g
)
.
(43)
However, these altered gravitoelectromagnetic equations give ris
e to an additional
Lorentz-force-like term,
√
Gρ
g
~
E
g
+
1
c
~v
g
×
~
B
g
, in the conservation of momentum
equation (
39
) so that now the theory violates conservation of momentum with or
without
29
Jefimenko
(
[2000]
,
[2006]
) considers making such a modification, though he focuses pri
marily on
gravitostatics (Newtonian gravity).
19
the troublesome factor of 4 in (
42
). One could imagine making further alterations to
the laws of gravitoelectromagnetism in order to alleviate all such pro
blems, but I will
not explore that avenue here.
30
In gravitoelectromagnetism the total mass of a planet, including th
e gravitational
field surrounding it, is less than the mass of the matter that compos
es the planet alone.
This is somewhat strange. But, it is a strangeness that we should ge
t used to: it will
remain in general relativity
31
and was already present (for energy but not mass) in
Newtonian gravity. Because it is this total mass
ρ
m
+
ρ
g
, and not
ρ
m
alone, which acts
as the source of gravitation in (
41
), the gravitational field far away from a planet will
be a bit weaker than you’d expect from (
11
) or (
27
). This effect is generally small. For
example, the mass of the earth’s gravitational field is about
−
4
.
2
×
10
−
10
times the mass
of the earth itself.
32
5 The Mass of the Gravitational Field in General
Relativity
The theory of general relativity is normally presented as a theory o
f spacetime geometry
in which gravity is represented by the curvature of spacetime and n
ot by a field on
spacetime. Such a geometric approach to general relativity makes
it difficult to pose
the questions we’ve been asking of the previous theories as it appea
rs,
prima facie
,
that there is no gravitational field to ponder the mass of. One might
argue that even
though there is no separate field defined on spacetime, there exist
s some aspect of the
spacetime which deserves the name ‘gravitational field’, such as the
connection Γ
α
βγ
, the
Riemann tensor
R
α
βγδ
, or the metric
g
μν
.
33
However, a field which is part of spacetime
looks quite different from the fields on spacetime that we have been e
valuating in the
preceding sections.
In order to more easily build on what we’ve learned so far, it will be helpf
ul to
focus on an alternative formulation of general relativity in which gra
vity is treated as
a field on spacetime. I will call this the field-theoretic approach to co
ntrast it with the
geometric approach described above.
34
The field-theoretic approach to general relativity
30
See the references in footnote
35
for more on this sort of problem as it arises in the context of
general relativity (in particular,
Misner
et al.
,
[1973]
, p. 186).
31
This effect is mentioned often in discussions of general rela
tivity, e.g., (
Arnowitt
et al.
,
[1960]
;
Weinberg
,
[1972]
, sec. 3.1, 7.6, and 8.2;
Misner
et al.
,
[1973]
, p. 467;
Ohanian
,
[unpublished]
).
32
Weinberg
(
[1972]
, p. 70) gives an estimate of this contribution which appears
to be off by a factor
of two.
33
The merits of these options are assessed in (
Lehmkuhl
,
[2008]
, sec. 4) and references therein.
34
Finding appropriate terminology here is difficult as there ar
e a range of subtly different views about
the status of spacetime geometry and gravitational field in g
eneral relativity.
Lehmkuhl
(
[2008]
) carves
things up differently. According to Lehmkuhl, a ‘field interp
retation’ of general relativity ‘claims that
the geometry of spacetime can be reduced to the behavior of gr
avitational fields’. This category is
meant to include the field-theoretic approach alongside oth
ers. On the field-theoretic approach, the flat
background spacetime is of course independent of the gravit
ational field and not reduced to it. But, the
metric
g
μν
—which is interpreted as specifying the geometry of spaceti
me on the geometric approach—is
straightforwardly determined by the gravitational field ac
cording to (
44
).
20