arXiv:0807.2838v1 [gr-qc] 17 Jul 2008
A Bi-Metric Theory with Exchange Symmetry
S. Hossenfelder
∗
Perimeter Institute for Theoretical Physics
31 Caroline St. N, Waterloo Ontario, N2L 2Y5, Canada
(Dated: July 17, 2008)
We propose an extension of General Relativity with two different metrics. To each metric we define a LeviCevita connection and a curvature tensor. We then consider two types of fields, each of which moves according
to one of the metrics and its connection. To obtain the field equations for the second metric we impose an
exchange symmetry on the action. As a consequence of this ansatz, additional source terms for Einstein’s field
equations are generated. We discuss the properties of these additional fields, and consider the examples of the
Schwarzschild solution, and the Friedmann-Robertson-Walker metric.
During the last decades, experimental achievements in astrophysics provided us with new insights about our universe.
The more precise our observations have become, the more
obvious also the insufficiency of our understanding have become. Today’s research in cosmology is accompanied by the
group of cosmological problems, which strongly indicate that
our knowledge about the universe is incomplete. Most importantly, it is microscopic explanations for dark matter and dark
energy that we are lacking. A lot of effort has been invested
into studies of fields with unusual equations of states which
can account for one or the other constituent.
In this paper we propose an extension of General Relativity with fields that experience space-time to have a metric h
different from our usual one
g. This causes additional source
terms to Einstein’s field equations that have properties unlike
those of our standard matter. The new fields do not move
according to the Levi-Cevita connection, but according to a
non-metric, torsion-free connection derived from the second
metric
h. To obtain the equations of motion for the second
metric, we propose a symmetry between both types of matter
and the according metrics.
Different versions of bi-metric theories and their potential
importance to explaining observational evidence have previously been investigated in [1, 2, 3, 4], and the approaches in
references [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
study similar symmetry considerations.
This paper is organized as follows: The general setup with
the two metrics and two types of fields is introduced in sectio
n
I. In the section II, we define the connections, and in section
III we construct the action for the new sort of fields and couple it to General Relativity (GR). In section IV we use the
exchange symmetry to obtain the complete set of equations
including those for the second metric, and then investigate the
example of the Schwarzschild solution, and the FriedmannRobertson-Walker metric in V and VI. After a discussion of
the scenario and its possible observable consequences in VII,
we conclude in section VIII.
Throughout this paper we use the convention
c
=
h¯
= 1. The
signature of the metric is
(
−
1
,
1
,
1
,
1
). Small Greek indices are
space-time indices and run from 0 to 3. ∗Electronic address: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>
I. A BI-METRIC THEORY
We consider a bi-metric theory with metrics
g and
h of
Lorentzian signature that define two different ways of measuring angles, distances and volumes on a manifold M. Changing
from
g to
h changes the map from the tangential space, TM, to
the co-tangential space, TM∗
. Since we then have two ways to
raise and lower indices, we will use a notation with two types
of coordinate-indices so we do not spoil the Ricci-calculus
.
For this, we will denote coordinate indices that are raised and
lowered by
h with underlines. In case objects do not carry
indices we underline them in total.
We will further introduce two sorts of matter on
M: one
that moves according to the usual[22] metric
g and the measure it implies, the other one that moves according to the other
metric
h. We will refer to these fields as
g-fields and
h-fields,
respectively; the equations of motions will be specified in section III. Related, we consider two types of observers on our
spacetime, the one made up of
h-fields who measures with the
metric
h, the other one made of
g-fields who measures with
metric
g. They will have to set their observations in relation to
each other in a consistent way, much like different observer
s
in Special Relativity.
Such, we have on the one hand the
h-observer who sees a
g-field with unusual behavior, and on the other hand the
g
-
observer who thinks this field to be perfectly normal matter.
The same situation applies for
g and
h exchanged. To take this
into account we will consider a map
Ph, which is an automorphism on the tensor-bundle, and which maps h-fields as the h-observer sees them to h-fields as the g-observer sees them.
Similarly, we have a map
Pg, which maps
g-fields as the
g
-
observer sees them to
g fields as the
h-observer sees them.
These maps conserve the tensor structure of objects, i.e. a
tensor of rank
(
r,
s
) is mapped to a tensor of rank
(
r,
s
), and
they are linear in the field’s components. Most importantly,
they assign a two-tensor
hκν to the metric
h, which we will
denote by
h, and a two-tensor
gκν to the metric
g, which will
be denoted by
g
:
g
=
Pg
(
g
)
,
h
=
Ph
(
h
)
. (1)
Since these maps are linear, we can write in components Phνν Phκκ hνκ = hνκ (2) [Pg]νν [Pg]κκ gνκ = gνκ . (3)
2
We will in the following refer to these maps as ‘pull-overs’.
One has to be careful however when pulling elements over
from the one observer to the other. Since the components of
pulled over tensors are mapped from the tangential into the
cotangential space with a different metric, the notation of the
inverse does no longer match with the use of indices. I.e. the
pull-over P
−1
h
of h
νκ
is not hκνg
κεg
νλ (where hκν is the pullover of hκν) but rather a tensor h
−1 with the property that
[h
−1
]
ελhεκ = δ
λ
κ
. We will try to avoid this by not using the
pull-overs as long as not absolutely necessary (and we will
see that we can rather successfully do so).
The introduction of these pull-overs is an assumption. Its
usefulness lies in enabling us to now chose different coordinate systems for the g and h-observer. We could for example apply a change of coordinates only on the h metric. This
would imply an according change in the pull-over (a multiplication with the inverse of the coordinate transformation),
but it would not make it necessary to also change the coordinate system the g-observer has chosen for his description.
The most obvious choice for the pull-over would just be the
identity. We will however see later that we can not fix these
pull-overs, but that they have to be determined from the field
equations, and can not in general be chosen to be the identity.
We further define a map a that transforms the one metric
into the pull-over of the other
gελ = a
ν
ε a
κ
λ
hνκ . (4)
Since both g and h are symmetric, a is not completely determined by (4). We fix the remaining six degrees of freedom
(dof) by requiring it to be symmetric, i.e. g
κνa
ε
ν = a
εκ = a
κε
.
We can pull over a by
a
ν
ε = [Pg]
ε
ε
a
ν
ε

Ph
ν
ν
, (5)
which then gives the relation
gελ = a
ν
ε a
κ
λ
hνκ . (6)
This pulled over quantity is also required to be symmetric. It is
further useful to define a combination of a and the pull-overs
that maps g to h via
a
ν
ε = a
ν
ε
[Ph]
ν
ν , (7)
gελ = a
ν
ε a
κ
λ
hνκ . (8)
And by raising and lowering some indices we also have
gελ = a
ν
ε aλν , hνκ = a
ε
ν
aεκ . (9)
In this formulation, the introduced map a is a convenience
and not a dynamical field, since it is defined by relating g to
Ph(h). The dynamical quantities are g and h, as well as the
both pull-overs Ph and Pg, and possible additional matter and
gauge fields. That means for the variation we have to keep a
fixed. We specify the properties of the field under variation by
requiring
δa
νκ = 0 . (10)
We will see later that this requirement makes for an interesting
scenario as it results in quite unusual properties of the h-fields.
One can read this off already from Eq. (4). Demanding (10)
to hold will imply that it is the inverse of the second metric
that behaves under variation like the usual metric. Unlike the
problems that one would run into by just using the inverse
metric as a second metric (because one had to treat a covariant
field as a contravariant one) the approach proposed here by
introducing a field that only behaves under variation as the
inverse metric, is manifestly covariant.
II. CONNECTIONS
To g one can define a Levi-Cevita connection in the usual
way that we will denote as (g)∇. Similarly, one can define a
Levi-Cevita connection to h, and we will denote this connection as (h)∇. To both metrics with their connections, one can
construct the curvature tensor, the Ricci tensor, and the curvature scalar, that we will denote as (g)R and (h)R, respectively.
In detail one has
(g)Γ
ε
νκ = g
εα(g)Γανκ , (11)
(g)Γανκ =
1
2
(∂νgακ +∂κgαν −∂αgνκ) , (12)
(h)Γ
ε
νκ = h
εα(h)Γανκ
, (13)
(h)Γανκ =
1
2
(∂νhακ +∂κhαν −∂αhνκ) . (14)
We can further define a pulled over derivative for the h-field,
which we denote as (h)∇, by requiring it to be a torsion-free
but non-metric connection that preserves h. From (h)∇h = 0
we then find the connection coefficients to (h)∇ to be
(h)Γ
ε
νκ =
1
2
[Ph(h)]εα (∂νhακ +∂κhαν −∂αhνκ) . (15)
As mentioned previously, one should keep in mind that
[Ph(h)]εα is not h
εα but the inverse of hεκ, and further that
the indices on (h)Γ are lowered and raised with g. From the
above one also finds for h, the determinant of hνκ,
∂κh = 2h
(h)Γ
ν
νκ . (16)
One can do a similar construction to pull over (g)∇ to obtain
a derivative (g)∇, which is not metric with respect to h but
preserves g, i.e. (g)∇g = 0. This construction of derivatives
now puts a further requirement on the pull-overs because they
have to be compatible with the tensor structure. So, for an
arbitrary tensor A we then have
Ph(
(h)∇A) = (h)∇Ph(A) , (17)
Pg(
(g)∇A) = (g)∇Pg(A) . (18)
In components we had e.g. for A
λ
κε
[Ph(
(h)∇νA
λ
κε)]λ
νκε =
(h)∇ν[Ph(A
λ
κε)]λ
κε . (19)
3
Since both connections as well as their pull-overs have to
be torsion free, this implies the pull-overs have to be integrable and are generated by two vector fields v and w such that
[Pg]
ν
κ ≡ Wν
κ = ∂κw
ν
, and [Ph]
ν
κ ≡V
ν
κ = ∂κv
ν
. The pull-overs
thus carry four dof each, after requiring them to be torsion free
and metric-compatible in the above described fashion.
However, we will for the variation not assume these requirements are already fulfilled, as this is in conflict with the
dof we need. As explained previously, the independent variables are g and h, each of which has ten dof. Since the a’s
and a’s do not carry degrees of freedom, the pull-overs need
to carry these ten dof since a conjunction of both, the a’s and
the pull-overs, relates g to h, as can be read off from Eq. (8).
Therefore, prior to the variation we can only assume the connections are torsion free, and take into account metric compatibility after the variation. For the usual connections, the variation over the connection together with the torsion-free-ness
implies metric compatibility as usual. For the additional connections, we will here not explicitly add a term to the action
to generate it but subsequently assume metric-compatibility
since it seems to be a desirable feature (though the scenario
could be considered in more generality).
To summarize this section: Each metric defines its own
Levi-Cevita connection, and after pulling them over these induce two non-metric connections, which will describe the motion the g-observer assigns to the h-fields and vice versa.
III. AND ACTION
Now let us add some physics. Consider we have an h-field
that behaves not according to the usual Levi-Cevita connection, but according to the connection metric with respect to h,
a field that feels angles and distances as defined by h not g.
For a massless scalar h-field φ the action that gives the equations of motions as the g-observer sees them could look like
S =
Z
d
4
x
√
−h Ph

h
νκ (h)∇κφ
(h)∇νφ

, (20)
where h = det(Ph(h)) so we have pulled over the determinant
and the measure is appropriately invariant.
For a scalar field the covariant derivative is of course just
the partial one so it does not matter according to which metric the connection is metric, but in general this will not be the
case. One can construct Lagrangians L for other types of hfields than scalars in a similar way by replacing the usual metric with the other one, and the usual Levi-Cevita connection
with the one belonging to the other metric, and then pulling
over. If it was just to make a scalar, one could consider the
density weight to be √
√ −g. However, the relevance of putting
−h instead becomes apparent when one takes the variation
over the field and its connection to obtain the equations of motion (eom). In order to convert the pull-over of a term (h)∇
νAν
into a total derivative the prefactor needs to be √
−h not √
−g,
so it is compatible with the derivation used in the Lagrangian.
For such a term then to vanish one uses equation (16), which
guarantees the validity of Gauss’s law. The resulting eom are
then just
Ph

(h)∇
α(h)∇αφ

= 0 , (21)
which is by definition of the pull-over identical to
(h)∇
α(h)∇αPh(φ) = 0 . (22)
Since the pull-over is invertible, the eom (21) are also equivalent to
(h)∇
α(h)∇αφ = 0 , (23)
which are the eom the h-observer would expect. Now we add
such a field to GR:
S =
Z
d
4
x
√
−g

(g)R/8πG+L

+
√
−h Ph(L ) . (24)
Upon variation, the first two terms give just the standard, and
the last term yields
Z
d
4
x
δ
δhνκ
√
−h Ph(L )

δhνκ , (25)
which we have to rewrite into a variation over gκν so we can
add the terms. We vary (see also Appendix A)
gελhκνa
εκa
µν = δ
µ
λ
(26)
and rewrite into
δhκλ = −[a
−1
]
µ
κ[a
−1
]
ν
λ
δgµν , (27)
where a
−1
is the inverse of a
[a
−1
]
κ
β
a
β
ν = δ
κ
ν
. (28)
We put in some pull-overs and their inverse, and rewrite into
a
κ
κ to make the symmetry more apparent. Then, we can add
all terms together and obtain from the variation (see Appendix
A) of the action the equations
(g)Rκν −
1
2
gκν
(g)R = 8πG

Tκν −
s
h
g
a
ν
ν a
κ
κ Tνκ!
(29)
with the sources
Tµν = −
1
√
−g
δL
δg
µν
+
1
2
gµνL (30)
Tνκ = −
1
√
−h
δL
δh
νκ
+
1
2
hνκL
=

P
−1
h

−
1
√
−h
δL
δh
νκ
+
1
2
hνκL

νκ
, (31)
where the last line is only to clarify how the rewriting of the
variation from h to h comes into play.
This change of sign we see appearing here in equation (27)
does occur only for the gravitational stress-energy-tensor, i.e.
4
the source term to Einstein’s field equations since it is a consequence of taking the variation with respect to the metric. It
does not occur if one derives the kinetic energy momentum
tensor (via Noether’s theorem), neither does the Lagrangian
have a negative kinetic energy term. Since then the sum as
well as the difference of both stress-energy-tensors is conserved, this means they are separately conserved; both fields
interact only gravitationally. Thus, despite the presence of
negative gravitational masses, there is no vacuum instability
because the kinetic energy of both sorts of fields remains positive, is conserved as usual, and a pair production of negative
and positive gravitational masses out of vacuum is not possible.
IV. WITH EXCHANGE SYMMETRY
In the previous section we have only considered the perspective of the g-observer and the field equations for the g
metric. We have however used that the fields g and h are not
independent. For symmetry reasons, the independent variables should be g and h, as well as the two pull-overs, with
which we further obtain g and h via (1). h is then however
related to g via eq. (4), and g to h via eq. (6).
Based on this consideration, we add matter fields to GR: a
g-field ψ, and an h-field φ, and request the action be symmetric
under exchange of g with h, and exchange of g-fields with hfields. This way we obtain
S =
Z
d
4
x
√
−g

(g)R/8πG+L (ψ)

+
√
−hPh(L (φ))
+
Z
d
4
x
p
−h

(h)R/8πG+L (φ)

+
p−gPg(L (ψ))(32),
where the first two terms are varied with respect to g using
eq. (4) as done in the previous section, the last two terms with
respect to h using eq. (6), and one should keep in mind that
h = det(Ph(h)) 6= h and g = det(Pg(g)) 6= g. The eom for the
matter fields are the usual ones and their pull-overs, and the
missing field equations for the second metric take the form
(h)Rνκ −
1
2
hνκ
(h)R = 8πG

Tνκ −
rg
h
a
κ
κa
ν
νTκν
, (33)
with the previously defined stress-energy tensors from eqs.
(30) and (31). Since equation (29) contains h rather than h
and equation (33) contains g rather than g, the pull-overs are
necessary ingredients. We can make this more apparent by
explicitly putting them into the equations:
(g)Rκν −
1
2
gκν
(g)R = Tκν −V
s
h
g
a
ν
ν a
κ
κ Tνκ (34)
(h)Rνκ −
1
2
hνκ
(h)R = Tνκ −W
rg
h
a
κ
κ
a
ν
νTκν , (35)
where V is the determinant of Ph, and W is the determinant of
Pg. (Or, to be more precise their absolute values since the volume element is positive.) Note that these equations would not
be invariant under coordinate transformations for each of the
observers separately without the pull-overs, since the factor
h/g was not an invariant in this case.
The virtue of doing this is that we can now chose the coordinate systems for g and h separately which seems to be natural.
For example, there is no reason to expect that the coordinate
system that one observer would consider ‘free-falling’ will
agree with the other one’s, since they both move according to
different connections. Thus it will not in general be clear what
a gauge condition on the one metric does to the other one, and
it seems more useful to export this lack of knowledge into the
pull-overs.
The field equations (34) and (35) need to fulfill the contracted Bianchi identities with respect to the matching LeviCevita connection. Since the stress-energy tensor of the gfield is already covariantly conserved with respect to (g)∇,
and the stress-energy tensor of the g-field is similarly covariantly conserved with respect to (h)∇, this leaves us with each
four equations for the additional source terms constituted of
the fields behaving with respect to the non-metric connection.
After assuming metric-compatibility of the pulled-over connections as explained in section II, these each four equations
constrain the remaining four dof in the pull-overs that appear
in these terms.
One should note however that the pull-overs will need additional initial conditions, and unless one imposes further symmetry requirements esp. their determinants can be multiplied
by an arbitrary constant. One can also interpret this as there
a priori being no way of telling whether the coupling between
the two types of fields and gravity is equally strong or has an
additional pre-factor. However, instead of introducing additional coupling constants we will leave the constants in the
pull-overs and treat them as parameters of the model that, ideally, have to be determined by observational constraints. Eqs.
(34) reduce to the standard field equations in the limit where
the energy density of the h-fields is very small and/or the determinant of the pull-over is small such that the coupling is
very weak.
To summarize this section: We have 10 components for
each g and h. Equations (34) and (35) provide each 10 equations that are related by the (contracted) Bianchi identities.
These two times 4 equations fix the two times 4 degrees of
freedom left in Pg and Ph after requiring metric-compatibility
of the pulled-over connections, which leaves us as usual with
4 degrees of freedom to chose the coordinate systems for each
metric.
V. EXAMPLE I: THE SCHWARZSCHILD METRIC
To obtain a better understanding of the workings, we consider the case with T ≡ 0, and with only a spherically symmetric source of usual g-fields outside of which there is vacuum
and we thus have the Schwarzschild solution for g. This solution has one free constant, M, that is the integral over the
energy density of the source. Making the obvious ansatz of
spherical symmetry for h in the same coordinate system, we
find also a Schwarzschild solution with one free constant to
5
be fixed by integrating over the source term, and the pull-over
can be set to be constant. Since g/h = 1 , we see that the
integrational constant is just −McW < 0, where cW is the determinant of the pull-over. A further symmetry requirement
that the asymptotic limit of h be just the Minkowski metric η,
as is that of h and g, fixes the pull-over to be the identity and
cW = 1. We then have
gtt = −

1−
2M
r

, grr = −1/gtt ,
gθθ = r
2
, gφφ = r
2
sin2
θ , (36)
htt = −

1+
2M
r

, hrr = −1/htt ,
hθθ = r
2
, hφφ = r
2
sin2
θ , (37)
and hκν = hκν.
One can now compute the connection coefficients according to eq. (15) and obtain the geodesic equations for an
h-field in this background of a g-source. Since h is just a
Schwarzschild metric with a negative source one sees e.g. by
taking the Newtonian limit that an h-particle will be repelled
by the g-source. It should be emphasized that in the here presented approach this does not result in contradictions as those
pointed out by Bondi [5] since the h-fields do move according
to a different connection and thus the equivalence principle
does not apply (see also Appendix B).
In case the radius of the matter source fell below its
Schwarzschild radius, and we had a black hole geometry for
g, the metric h would not have a horizon (the h-fields are repelled by the source). Note also that this evident symmetry of
the above metrics is not as obvious in every coordinate system. For example changing to one of the more well-behaved
systems with e.g. in/out-going EF-coordinates will be a nice
transformation for the usual metric g, but completely mess up
the other metric h. The reason is just that for the h-observer
in/out-going means something different.
As this example shows, the bi-metric model is not causal
in the sense that the h-field’s propagation does not need to lie
within the lightcone of the standard fields. The h-metric on
the manifold thus describes a global causal structure that in
general will be different from that of the g-matter. Because of
the symmetry between both however, the h-field’s propagation
is causal as well if the properties of a curve’s tangent vector
are defined through the respective metric. Physically, there
are then two ways to form closed timelike curves: The one is
through an interaction between both types of matter which together could carry information around a closed timelike curve.
Since the interaction between both types of matter is mediated
only by gravity and very weak, this would be relevant only for
curves going through regions where the gravitational interaction is strong. The other possibility is through a space-time
structure that allows closed timelike curves, which is known to
be possible in the presence of negative energy densities [19],
even for only small amounts [20]. It remains an open question though whether these solutions actually describe natural
settings.
It is further worthwhile to point out that the analysis
about stability of the negative mass Schwarzschild black hole
worked out in [21] does not apply to the here discussed case,
since a perturbation of the metric h is related to that of the
usual metric g via the field equations. Since g is stable as
usual, there is no reason to expect any instabilities for the metric that the h-observer would measure.
Both types of fields only interact gravitationally, so the hfields constitute a kind of very weakly interacting dark matter.
Since both kinds of matter repel, one would expect the amount
of h-matter in our vicinity to presently be very small.
VI. EXAMPLE II: FRIEDMANN-ROBERTSON-WALKER
We have the usual Friedmann-Robertson-Walker (FRW)
metric for g
ds2 = −dt2 +
a
2
1−kr(dr2 +dΩ
2
) , (38)
and make the ansatz for h
ds2 = −dt2 +
b
2
1−kr(dr2 +dΩ
2
) , (39)
where k = −1,0,+1. To preserve the symmetry of the FRW
metric, we further expect the pull-overs to only act on the time
coordinate and we have a
2/(1 − kr) = [Pg(g)]rr and b
2/(1 −
kr) = [Ph(h)]rr. For the sources, we use the notation
T
0
0 = ρ , T
i
i = p (40)
T
0
0 = ρ , T
i
i = p . (41)
(Here, the indices i,i = 1,2,3, and are not summed over). The
first Friedmann equations for both metrics then are

a˙
a
2
= ρ−W

b
a
3
ρ−
k
a
2
, (42)

b˙
b
2
= ρ−V

a
b
3
ρ−
k
b
2
, (43)
where a dot indicates a derivative with respect to t, and the
conservation laws read
∂t(ρ−W

b
a
3
ρ)+3
a˙
a

ρ+ p+W

b
a
3
(ρ+ p)
!
= 0 (44) ,
∂t(ρ−V

a
b
3
ρ)+3
b˙
b

ρ+ p+V

a
b
3
(ρ+ p)
!
= 0 (45) .
Since the pull-overs here act only on the time-coordinate, the
solutions for a and b together with the determinants V and
W then give h and g. For example if one wanted to compute
the equation of motion for an h-photon, one had htt = −W2
and hrr = b
2/(1−kr) with which one obtains the non-metric
connection via eqs. (15).
We will in this present work not attempt to discuss the solutions of these equations in all generality, but instead we consider some specific cases of interest. In the following, cV and
cW are positive valued constants.
6
1. In case ρ = 0, W does not appear in the equations. If ρ
is a matter field, i.e. p = 0, then V = cV is a solution; if
ρ is a radiation field, i.e. ρ = 1/3p, then V = cV a/b is
a solution. In this case the V could be absorbed into the
metrics (e.g. by changing both into comoving coordinates). One should note that with this choice of sources
eq. (43) does not have a solution for k = 0,1. The case
for ρ = 0 is similar.
2. If both ρ and ρ are matter fields then V = cV , W = cW
is a solution.
3. If both ρ and ρ are radiation fields then V = cV , W = cW
is also a solution.
4. If ρ is a matter field and ρ is a radiation field then W =
cW b/a and V = cV . Similarly, if ρ is a matter field and
ρ is a radiation field then V = cV a/b and W = cW . In
these cases, it is not possible to set both W and V to be
constant.
5. If both sources are cosmological constants we have
W = cW (b/a)
3
and V = cV (a/b)
3
. Note that for certain
values of the constants cW and cV the curvature needs
to be negative.
As previously mentioned, the pull-overs are only determined
up to constants that have to be specified in the initial conditions. With a suitable choice of these constants, one can
achieve the additional source term to be negligible. Though
this scenario does not seem particularly compelling, we want
to point out that for this reason it is possible to reproduce standard GR up to small corrections.
VII. CONSEQUENCES AND POSSIBLE OBSERVABLES
In the previous sections we have studied an extension of
GR in whose framework sources with negative gravitational
energy appear in the field equations. These additional h-fields
interact only gravitationally with our standard matter, and thus
couple only extremely weakly. In this section we want to mention some reasons why this scenario is interesting and worth
further examination.
The model we laid out is purely classical. Nevertheless it
is worthwhile to consider the vacuum expectation value of the
stress-energy tensor for quantum fields that are coupled to the
classical background. We will assume that the field content
for both, the g-fields and the h-fields, is identical such that we
have e.g. two copies of the Standard Model. The vacuum expectation value of these quantum fields is just proportional to
the respective metric. Though the constant of proportionality
is technically seen divergent, one expects this vacuum energy
to be regularized at the Planck scale mp. This leads one to
the well known problem that this vacuum energy density is
∼ m
4
p
, and far too large to ever allow our universe to form the
structures we observe.
If we consider the vacuum solution in the model with exchange symmetry however, we expect a symmetry between
both metrics. In the case with the maximal number of spacetime symmetries, both would just be the Minkowski metric.
We then have h/g = 1, and the pull-overs are just the identity. Since the matter content of both types of fields is identical, this means the source terms in eqs. (34) and (35) cancel
identically, no matter how large their values are. This is a
consequence of the additional symmetry. Whether or not this
solution is stable or would run away if the constants did not
exactly cancel requires further investigation. Needless to say,
the measured value of the cosmological constant is not zero,
but at least it is closer to zero than to m
4
p
.
This bring us to another point to be mentioned, namely the
extraction of observables from the data, e.g. the high-redshift
supernovae data or the WMAP results. Underlying the data
analysis to obtain constraints on the parameters in the ΛCDMmodel is the usual GR formalism. Unfortunately, some parts
of this formalism can not be applied in the model discussed
here. For example, to use cosmological perturbation theory a
relevant parameter is the relative size of perturbations δρ/ρ.
Typically, one infers from the CMB measurements the perturbations at freeze-out were small ≈ 10−5
. This however
is the size of perturbations relative to the observable (usual)
matter density. Since we now have an only gravitationally interacting density contribution that is negative, and one further
would hope for symmetry reasons that both densities are of
the same order of magnitude, the total gravitating density can
be smaller than the observed one. Then, the relative density
fluctuations could be larger. Besides this, both components of
matter repel each other which is an effect usually not present.
Another feature of the scenario becomes clear from the previously discussed example of the Schwarzschild metric. If
there was a localized source of negative energy, it would act
as a gravitational lens - but unlike usual matter this would be a
diverging lens since it would repel our (usual) photons. Such
a lensing event would typically lower the luminosity of the
source, an effect that could potentially add up over distance if
the distribution of such sources is substantial. The detection
of a diffractive lensing event could serve as a smoking gun
signal for the here proposed scenario.
VIII. SUMMARY
We have studied an extension of General Relativity with
two metrics, and two sorts of fields. Each field moves according to the Levi-Cevita connection of one of the metrics. The
new sort of fields only interacts gravitationally with our usual
matter. We have coupled these fields to General Relativity.
By requiring the action to be symmetric under exchange of
the two metrics, and their fields, we obtained a model from
which we could derive the equations of motions for the two
sorts of fields, as well as the field equations for both metrics.
It turned out that the additional fields can make a contribution
to the gravitational stress-energy tensor with a negative energy
density. We argued that this does not imply a vacuum instability since the kinetic energies are still strictly positive and
conserved. We further investigated the spherical symmetric
example with a source of usual matter, and we found that the
7
newly introduced particles would be repelled by this source.
We also derived the Friedmann-equations within this scenario
and discussed some general properties of possible solutions.
Finally, we mentioned some possible consequences for observables, most importantly a diffractive gravitational lensing
effect.
We hope to have shown that the here proposed bi-metric
model with exchange symmetry has interesting properties,
and that it can potentially shed light on some so far unresolved
questions in cosmology and astrophysics.
Acknowledgements
I thank Lee Smolin for helpful discussions, and for reading
various drafts of this paper. Research at Perimeter Institute for
Theoretical Physics is supported in part by the Government
of Canada through NSERC and by the Province of Ontario
through MRI.
Appendix A
Contracting Eq(4)
gµν = a
α
µ a
β
ν hαβ (46)
with g
µκ yields
δ
κ
ν = a
καa
β
ν hαβ = a
καa
λβ gλνhαβ . (47)
Taking the variation with use of (10) one obtains
a
καa
λβ gλνδhαβ +a
καa
λβ hαβδgλν = 0 (48)
⇔ a
καa
λβ gλνδhαβ +g
κλδgλν = 0 , (49)
and after contracting with gκµ
δgµν = −a
α
µ a
β
ν δhαβ . (50)
From (46) one reads off that the inverse of a
κ
ν
is
[a
−1
]
κ
β = a
καhαβ , (51)
where
[a
−1
]
κ
β
a
β
ν = δ
κ
ν
. (52)
Since aκν is symmetric, so is [a
−1
]κν and we also have
[a
−1
]
β
κa
ν
β = δ
ν
κ
. (53)
We then use (53) to bring the a’s in (50) to the other side
δhκλ = −[a
−1
]
µ
κ[a
−1
]
ν
λ
δgµν . (54)
If we consider pulling over one of the indices on a
−1 with use
of Ph, we obtain
[a
−1
]
κ
ν
[Ph]
κ
κ = a
καhαν[Ph]
κ
κ . (55)
By putting in a pull-over for the index α and its inverse, and
absorbing the pull-overs in the definition (7) and (8) we get
[a
−1
]
κ
ν
[Ph]
κ
κ = a
µα
hακ = a
µ
κ . (56)
The pull-overs are linear, so we have
δ
δh
κλ Ph(L ) = 
Ph

δL
δhκλ 
κλ
=
δL
δhκλ
[Ph]
κ
κ[Ph]
λ
λ
. (57)
Note that this is not a performance of the variation, but a
rewriting of the derivative with the aim to express the variation of L in a way that the symmetry becomes more apparent.
We could leave this term in the initial form where variation
is with respect to h
νκ but we want to make use of the fact
that the form of L is symmetric to the usual Lagrangian under exchange of gκν with hκν (not hκν) and exchange of the
respective covariant derivatives.
With Eq.(54) and (56) and we then obtain
δ
δh
κλ Ph(L )δh
κλ =

Ph

δL
δhκλ 
κλ
δh
κλ
=
δL
δhκλ
[Ph]
κ
κ[Ph]
λ
λ
δh
κλ
= −
δL
δhκλ
[Ph]
κ
κ[Ph]
λ
κ[a
−1
]
κ
µ
[a
−1
]
λ
ν
δg
µν
= −
δL
δhκλ
a
κ
µ
a
λ
ν δg
µν
, (58)
with which we return to Eq. (29).
Appendix B
If it wasn’t the case that a negative mass particle moved according to a different covariant derivative, one could construct
the following problem with negative sources in General Relativity: Gravity is a spin-two field. Thus, like charges attract
and unlike charges repel, and a negative mass particle should
be repelled by a positive mass source. On the other hand, the
negative mass particle moves according to the geodesic equation which does not know anything about the particle’s mass -
it only knows about the positive source background. Thus, the
negative mass particle should be attracted to the source as all
test particles. One would then be lead to conclude a negative
mass test particle was attracted and repelled likewise which
can be used to construct all kinds of nonsense.
The reason for this confusion is that the use of the usual
geodesic equation for the negative mass particle is inappropriate which one can understand most easily by interpreting the
covariant derivative as a coupling to the gravitational field that
conserves the total energy of the particle including the potential energy. For a negative mass test particle that is repelled
instead of attracted, the conservation law has to be different
since it couples differently to the background. This is similar
to the coupling of electrons and positrons to the electric field
being mediated by different covariant derivatives.
8
talk me about the pull over technic ,a map , and how the anti gravity emerge from exchange of symetrie