arXiv:0704.0228v1  [gr-qc]  2 Apr 2007
epl draft
Einstein vs Maxwell: Is gravitation a curvature of space,
a field in flat space, or both?
Theo M. Nieuwenhuizen
Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
PACS 04.20.Cv – Fundamental problems and general formalism
PACS 04.20.Fy – Canonical formalism, Lagrangians, and variational principles
PACS 98.80.Bp – Origin and formation of the Universe
Abstract. - Starting with a field theoretic approach in Minkowski space, the gravitational energy
momentum tensor is derived from the Einstein equations in a straightforward manner. This allows
to present them as acceleration tensor = const. × total energy momentum tensor. For flat space
cosmology the gravitational energy is negative and cancels the material energy. In the relativistic
theory of gravitation a bimetric coupling between the Riemann and Minkowski metrics breaks
general coordinate invariance. The case of a positive cosmological constant is considered. A
singularity free version of the Schwarzschild black hole is solved analytically. In the interior the
components of the metric tensor quickly die out, but do not change sign, leaving the role of time
as usual. For cosmology the ΛCDM model is covered, while there appears a form of inflation at
early times. Here both the total energy and the zero point energy vanish.
It is said that in introducing the general theory of rela-
tivity (GTR), Einstein made the step that Lorentz and
Poincaré had failed to make: to go from flat space to
curved space. Technically, this arises from the group of
general coordinate transformations [1, 2]. One fundamen-
tal difficulty is then how to deal with the physics of gravi-
tation itself, since there is only a quasi energy-momentum
tensor [3]. For gravitational wave detection, e.g., this
leaves open the question as to how energy can be faithfully
transferred from the wave to the detector. The proper en-
ergy momentum tensor of gravitation was derived only
recently by Babak and Grishchuk [4], who start with a
field theoretic approach to gravitation, in terms of a ten-
sor field hµν in a Minkowski background space-time. The
metric of the latter, ηµν = diag(1,−1,−1,−1), is denoted
in arbitrary coordinates by γµν = (γ
µν)−1. The Riemann
metric tensor gµν = (g
µν)−1, is then defined by
gµν = γµν + hµν ≡ kµν , g
det(gµν)
det(γµν)
. (1)
It is just a way to code the gravitational field, allowing
to expresses distances by ds2 = gµνdx
µdxν . Such a non-
linear way to code distances in a flat space is not uncom-
mon. For diffuse light transport through clouds, one may
express distances in the optical thickness, the number of
extinction lengths. If the cloud is not homogeneous, points
at the same physical distance are described by a different
optical distance and, vice versa.
The Maxwell view that gravitation is a field in flat
space, was actually the starting point for Einstein, and
reappeared regularly. Nathan Rosen [5], coauthor of
the Einstein-Podolsky-Rosen paper that led the basis for
quantum information, considers a bimetric theory, in-
volving the Minkowski metric and the Riemann metric.
Bimetrism is quite natural, with ηµν entering e.g. particle
physics, and gµν e.g. cosmology. Rosen considers covari-
ant derivatives Dµ of Minkowski space, with Christoffel
symbols γλ
·µν vanishing in Cartesian coordinates. When re-
placing in the Riemann Christoffel symbols partial deriva-
tives by Minkowski covariant ones,
·µν =
gλσ(∂µgνσ + ∂νgµσ − ∂σgµν) 7→
·µν =
gλσ(Dµgνσ +Dνgµσ −Dσgµν), (2)
the obtained Christoffel-type symbols Gλ
·µν are tensors in
Minkowski space. Inspired by the Landau-Lifshitz and
Babak-Grishchuk results, we may define the acceleration
tensor
Aµν =
DαDβ(k
µνkαβ − kµαkνβ), (3)
where kµν = γµν + hµν and in which the γγ terms do not
http://arxiv.org/abs/0704.0228v1
Th.M. Nieuwenhuizen
contribute. Then we can calculate the combination
τµν =
Aµν − (Rµν − 1
gµνR)
. (4)
In doing so, we make use of Rosen’s observation that Rµν
remain unchanged if one replaces all partial derivatives by
covariant ones in Minkowski space [5]. It appears that
all second order derivatives drop out from (4), leaving a
bilinear form in first order covariant derivatives,
τµν =
· · :λh
· · :ρ −
· · :λh
· · :ρ
hµλ:ρhν
·λ:ρ +
kµνhλρ:σhλσ:ρ −
hλρ:µhν
hµλ:ρh · · :νλρ +
hλρ:µh · · :νλρ −
kµνhλρ:σhλρ:σ +
kµνh ·ρ:λρ h
. (5)
in which X:µ ≡ DµX and raising (lowering) of indices
· · :ρ is performed with k
µν (kµν). τ
µν is a tensor
in Minkowski space. For Cartesian coordinates, it coin-
cides with the Landau-Lifshitz quasi-tensor. In general, it
coincides with the Babak-Grishchuk tensor γtµν/g. Inclu-
sion of matter is now much easier than in [4]. Inserting
the Einstein equations in the right hand side of (4), we
may write the Einstein equations in the Newton shape:
acceleration=mass−1×force,
Aµν =
Θµν ,
Θµν =
θµν , θµν ≡ τµν + T µν . (6)
Θµν is the total energy momentum tensor of gravitation
and matter. It is conserved, DνΘ
µν = 0, since Eq. (3) im-
plies DνA
µν = 0, because covariant Minkowski derivatives
commute.
As an application, let us consider cosmology, described
by the Friedman-Lemaitre-Robertson-Walker (FLRW)
metric,
ds2 = U(t)c2dt2 − V (t)
1− kr2
+ r2dΩ2
, (7)
dΩ2 = dθ2 + sin2 θdφ2.
Let us consider flat space, k = 0, and U = 1, V (t) = a2(t)
with a the scale factor. Then ds2 = c2dt2 − a2(t)dr2
is space-independent, implying that A00 = 0, due to the
shape (3). According to (6) it then follows that the total
energy density is zero, because the gravitational energy
density, τ00 = −3c4ȧ2/(8πGa2), is negative and cancels
the one of matter, T 00 = ρ, due to the Friedman equation.
In other words, such a universe contains no overall energy.
So far we have discussed an alternative, field theoretic
formulation of GTR. If we consider a local energy mo-
mentum density as a sine qua non property, then we are
led to consider Minkowski space as a fixed “pre-space”,
that exist already without matter, just as a region of
space ahead of the earth’s orbit is right now almost empty
(Minkowskian), and when the earth arrives, there will be
more gravitational and matter fields, but, in our view, no
change of space. Also for cosmology there is a different
interpretation. In GTR coordinates are fixed to clusters
of galaxies, this is called “coordinate space”, but due to
the increasing scale factor galaxies are said to move away
from each other: physical space (i.e. Riemann space) is
said to expand. Here we are led to another view: Coordi-
nate space is physical space, so clusters of galaxies do not
move away from each other in time. [6] However, the cos-
mic speed of light dr/dt = c/a(t), which was very large at
early times, keeps on decreasing, thus causing a redshift,
till a is infinite, when galaxies are invisible.
Relativistic Theory of Gravitation, RTG. Let us move
on to an extension of GTR, giving up general coordinate
invariance. Discarding a total derivative of the Hilbert-
Einstein action, Rosen expresses the gravitational action
d3xdt
−g LR in terms of [5]
c4gµν
·λσ −Gλ·µσGσ·νλ) =
128πG
× (2hµν:ρhµν:ρ − 4hµν:ρhµρ:ν − hν·ν:µh ·ρ:µρ ).
Involving only Minkowski covariant first order derivatives,
it is close to general approaches in field theory. Logunov
and coworkers continue on this [6]. The subgroup of gauge
transformations that transform hµν but leave coordinates
invariant, allows three extra terms [6],
Lg = LR − ρΛ +
ρbiγµνg
µν − ρ0
γ/g. (9)
Here ρΛ is the familiar energy related to a cosmological
constant. The ρ0 term describes a harmless shift of the
zero level of energy, δS = −
d3xdt
−γρ0. The bimetric
term ρbi couples the Minkowski and the Riemann metrics.
It acts like a mass term, because it breaks general coor-
dinate invariance, and has some analogy to a mass term
in massive electrodynamics. Logunov then imposes the
relation
ρΛ = ρbi = ρ0, (10)
which, in the absence of matter, keeps space flat, hµν = 0,
gµν = γµν and also Lg = 0. Thus one free parameter re-
mains. Logunov’s choice ρbi ≡ −m2c4/(16πG) < 0 leads
to an inverse length m and, in quantum language, a gravi-
ton mass h̄m/c. The negative cosmological constant can
be counteracted by an inflaton field [7]. The obtained
theory has some drawbacks, such as self-repulsive prop-
erties for matter falling onto a black hole, and a minimal
and a maximal size of the scale factor in cosmology [6] [7].
For a related approach to finite range gravity, based on a
generalized Fierz-Pauli coupling, see [8].
Einstein vs Maxwell
We shall focus on the opposite choice, a positive cosmo-
logical constant Λ, [9] [10]
Ωv,0H
9.78Gyr
ρbi ≡
= ρΛ. (11)
Now the graviton has an “imaginary mass”, m =
−2Λbi/c, it is a “tachyon”: Gravitational waves are
unstable at today’s Hubble scale. But this is of no con-
cern, since on that scale, not single gravitational waves
but the whole Universe matters, being unstable (expand-
ing) anyhow.
Though we take ρbi = ρΛ, Λbi = Λ, our further notation
is valid for the general case ρbi 6= ρΛ, Λbi 6= Λ.
The Einstein equations that couple the Riemann metric
to matter read
Rµν −
gµνR =
tot , (12)
tot = T
µν + ρΛg
µν + ρbiγρσ(g
µρgσν − 1
gµνgρσ).
Conservation of energy momentum, T
tot;ν = 0, imposes a
constraint due to the ρbi terms, [6]
= 0, or Dνh
µν = 0, (13)
which for Cartesian coordinates coincides with the GTR
harmonic condition ∂ν(
−ggµν) = 0 [2]. Thus the the-
ory automatically demands the harmonic constraint for
gµν , or, equivalently, the Lorentz gauge for hµν , thereby
severely reducing the gauge invariance of GTR.
Changes of Einstein’s GTR have mostly met deep trou-
bles with one or another established property, though not
all proposals are ruled out [1,11]. The present one is rather
subtle and promising. For most applications, the Hubble-
size ρΛ = ρbi terms in Eq. (11,12) are too small to be
relevant, so known results from general relativity can be
reproduced. Indeed, viewed from a GTR standpoint, Eq.
(13) is only a particular gauge, and actually often con-
sidered, while the cosmological constant only plays a role
in cosmology. Logunov checked a number of effects in
the solar system: deflection of light rays by the sun, the
delay of a radio signal, the shift of Mercury’s perihelion,
the precession of a gyroscope, and the gravitational shift
of spectral lines. [6] Likewise, we expect agreement for
binary pulsars. [11] Differences between GTR and RTG
may arise, though, for large gravitational fields, that we
consider now.
Black holes. It is known that true black holes, ob-
jects that have a horizon, do not occur in the RTG with
ρΛ, ρbi → 0. [5] But there are solutions very similar to it,
that might be named “grey holes”, but we just call them
“black holes”. The Minkowski line element in spherical
coordinates is simply γµνdx
µdxν = c2dt2 − dr2 − r2dΩ2.
The one of Riemann space is
ds2 = gµνdx
µdxν = U(r)c2dt2 − V (r)dr2 −W 2(r)dΩ2. (14)
In harmonic coordinates, the Schwarzschild black hole is
described by [2]
r − rh
r + rh
, Ws = r + rh, rh =
. (15)
The horizon radius rh equals half the Schwarzschild radius.
Let us scale r → rrh, and define
U = eu, V = ev, W = 2rhe
w, (16)
so that w is small near the horizon. The dimensionless
small parameter arising from ρbi = ρΛ, is very small,
λ̄ ≡ rh
2Λ = 2.38 10−23
. µ̄ ≡ rh
2Λbi = λ̄. (17)
The sum and difference of the (t, t) and (r, r) Einstein
equations give
ev−2w − w′(u′ − v′ + 4w′)− 2w′′
= ev(λ̄2 − 1
µ̄2r2e−2w) +
8πGr2h
ev(ρ− p), (18)
w′(u′ + v′ − 2w′)− 2w′′
µ̄2(ev−u − 1) + 8πGr
ev(ρ+ p), (19)
respectively. The harmonic condition imposes
u′ − v′ + 4w′ = r exp(v − 2w).
In the Schwarzschild black hole of GTR, there is no
matter outside the origin. We shall focus on that situation.
A parametric solution of these equations then reads
1 + η(eξ + ξ + log η + r0)
1− η(eξ + ξ + log η + r0)
, (20)
u = ξ + log η,
v = ξ − ln η − 2 log(eξ + 1), (21)
w = ηeξ + µ̄2(ξ + log η + w0).
where ξ is the running variable and η is a small scale.
Corrections of next order in η can be expressed in diloga-
rithms, but they are not needed since µ̄ is very small.
To fix the scale η, we note that energy momentum con-
servation implies, as in GTR, (ρ + p)u′ + 2p′ = 0. In
the stationary state all matter is located at the origin,
which is only possible if p(r) ≡ 0, implying ρ(r)u′(r) = 0.
This is obeyed for r 6= 0 since ρ = 0 there, but since
ρ(0) > 0 (it is infinite), we have to demand u′(0) = 0.
Let us define a factor α by α = µ̄2/η. The above solu-
tion brings w′(r) = ∂ξw/∂ξr = (e
ξ + α)/[2(eξ + 1)], so in
the interior w′ = 1
α. Since ev ≪ 1 there, Eqs. (18,19)
confirm that w′′ = 0, and with w(1) = O(η) this solves
Th.M. Nieuwenhuizen
1.510.5
Fig. 1: Black hole functions U(r), V (r) and W (r), scaled
by factors 10, (bold lines) for µ̄ = 0.1, compared to the
Schwarzschild solution (thin lines; the part V < 0 for r < 1 is
not shown). Inside the horizon, U and V decay very rapidly.
Since they remain positive, time keeps its role in the interior.
w(r) = 1
α (r − 1). Moreover, from the harmonic con-
straint (20) we have in the interior u(r)− v(r) + 4w(r) =
const = 2 ln η, implying that Eq. (19) yields in the interior
u′(r) = {exp[2α(r − 1)]− η2 − µ̄2}/(2η). From u′(0) = 0
we can now solve α,
α = log
, η =
ln 1/µ̄
. (22)
As seen in fig. 1, our solution (16,20,-22) coincides with
Schwarzschild’s for ξ ≫ 1. In the regime ξ = O(1), there
is a transition towards the interior ξ ≪ −1, where ex-
ponential corrections can be neglected. Both U = ηeξ
and V = eξ/η are very small there, but, contrary to the
Schwarzschild case, they remain positive: The behavior in
the interior of the RTG black hole is not qualitatively dif-
ferent from usual, be it that the gravitational field is large.
Width of the brick wall. The transition layer ξ = O(1)
acts like ’t Hooft’s brick wall, [12] of characteristic width
ℓ⋆ = ηrh. Comparing to the Planck length ℓP =
h̄G/c3,
we get
0.977 10−9
1 + 0.019 log(M/M⊙)
. (23)
If quantum physics sets in at the Planck scale, our ap-
proach makes sense only for M > 103 M⊙.
Motion of test particles. For RTG with a negative cos-
mological constant, [6] it was claimed that an incoming
spherical shell of matter is scattered off from a black hole,
a counter-intuitive finding. Let us reconsider this issue.
The motion of a test body occurs along a geodesic
+ Γµνρv
νvρ = 0, vµ =
. (24)
For spherical shells of in-falling matter one needs Γ001 =
U ′/(2U). This brings dt/ds = v0 = 1/(CiU), for some
Ci. Solving v
1 = dr/ds from gµνv
µvν = 1, we then get
dr/dt = (ds/dt)(dr/ds) = −c
U(1− C2i U)/V . We can
now fix Ci at the initial position r = ri, where the spherical
shell is assumed to have a speed dri/dt = vi = βic
Ui/Vi,
viz. Ci =
(1 − β2i )/Ui, with |βi| ≤ 1. The differential
proper time dτ =
Udt and length dℓ =
V dr bring in
the particle’s rest frame dℓ/dτ =
V/Udr/dt, yielding
1− U(r(τ))
U(ri)
(1 − β2i ). (25)
The extreme case is when βi = 0 at ri = ∞, dℓ/dτ =
1− U. To have |dℓ/dτ | < c, it thus suffices that 0 <
U ≤ 1, which is the case. Near the horizon, |dℓ/dτ | is
almost equal to c and the more the shell penetrates the
interior, the closer its speed gets to c. For an outside
observer, the time to see it hit the center of the hole,
dr/|ṙ| equals (rh/c)
dr exp[ 1
(v − u)]. It is finite
and predominantly comes from the horizon, T = rh/cµ̄
2.74× 1032M/M⊙ yr.
The approaches [6–8] have a similar a black hole. While
[8] properly has U ′(0) = 0, in Logunov’s case one has
µ̄2 < 0, so w′ = 1
α < 0 in the interior. This seems to
solve the paradox of “matter reflected by the black hole”:
In-falling matter just enters, but the Logunov coordinate
x = exp(w) − 1 is non-monotonic (x′ < 0 in the interior).
However, the situation is more severe: For α < 0, the
theory does not allow a solution with u′(0) = 0, depriving
that theory of a proper black hole. This condition can
neither be obeyed in GTR: If the central mass is slightly
smeared, the Schwarzschild black hole cannot obey energy-
momentum conservation in GTR.
Cosmology. Starting from the FLRW metric, the har-
monic condition brings two relations: U ∼ V 3 and k = 0:
Minkowski space filled homogeneously with matter re-
mains flat [6]. We may thus put U = a6(t)/a4
, V = a2(t).
Going from cosmic time t to conformal time τ =
a3a−2
yields the familiar Einstein equations, extended by Λbi
terms,
− Λbi
, (26)
= −4πG
(ρ+ 3p) +
− Λbi
The first is the modified Friedman equation, the second
corresponds to the first law d(ρtota
3) = −ptotda3 provided
we define ρtot = ρ+ρΛ+ρ2+ρ6 and ptot = p−ρΛ− 13ρ2+ρ6,
with ρ2 = −3ρbi/2a2 and ρ6 = ρbia4∗/2a6. Note that ρ2
acts as a positive curvature term.
The scale factor has an absolute meaning. If we as-
sume that a ≫ 1 and a ≫ a2/3∗ , Eq. (26) just coin-
cides with the ΛCDM model (cosmological constant plus
cold dark matter), that gives the best fit of the observa-
tions [9] [10]. The ρ2 term allows a positive curvature-
type contribution. At large times, there is the exponential
growth a(τ) = C exp(H∞τ) with H∞ = c
Λ/3. In cos-
mic time this reads a(t) = a
∗ [3H∞(t0 − t)]−1/3, where
Einstein vs Maxwell
t0 is “the end of time”, the moment where the scale factor
has become infinite. The minimal scale factor is zero: in
this theory a big bang can occur since ρbi > 0. With-
out including an inflaton field, Eq. (26) yields an initial
growth of the expansion a = (a2
3Λbi/2)
1/3. In cos-
mic time this reads a = a1 exp(ct
Λbi/6), i. e., a certain
inflation scenario starting at t = −∞.
Also in RTG the gravitational energy precisely compen-
sates the other energy contributions at all times. The
vacuum energy also vanishes: In empty space, the cosmo-
logical constant energy ρΛ cancels the ρbi terms, due to
Eq. (10). See Eq. (12) for gµν = γµν .
In conclusion, we have first written the Einstein equa-
tion in a form that involves the gravitational energy
momentum tensor. An underlying Minkowski space is
needed, in which gravitation is a field. The metric ten-
sor is a way to deal with it, but the equations for the field
itself exist too, see Eq. (6). For flat cosmology it follows
that the total energy vanishes.
Next we have broken general coordinate invariance by
going to the bimetric theory of Logunov, called Relativis-
tic Theory of Gravitation. We have shown that the choice
of a positive bimetric constant allows to regularize the
interior of the Schwarzschild black hole: time keeps its
standard role and escape is, in principle, possible. While
neither the Schwarzschild nor the Logunov black hole sur-
vives smearing of the central mass by a tiny pressure in the
equation of state, ours does. Our modification of the Ein-
stein equations involves the cosmological constant, so it is
of Hubble size, immaterial for solar problems. In cosmol-
ogy, the theory directly leads to the ΛCDM model, while
it could accommodate a positive curvature-like term. At
short times, there is a form of inflation. The gravitational
energy exactly compensates the material energy. The zero
point energy vanishes (“again”), though the cosmological
constant is finite and positive: It is canceled by the bimet-
ric terms.
Euclidean space, a special case of Riemann geometry,
seems to be invoked by Nature, at least far away from bod-
ies and in cosmology. Our approach supports the follow-
ing space-time interpretation: curvature is a geometric de-
scription of the gravitational field in flat space. Clusters of
galaxies do not move away from each other, but the speed
of light changes with cosmic time, dr/dt = [a(t)/a∗]
while the conformal speed is dr/dτ = c/a(τ) as usual.
An empirical way to establish the Minkowski metric is
to present the Einstein equations as (c4/8πG)Rµν −Tµν +
gµνT + ρΛgµν = ρbiγµν , and to measure the left hand
side, which in the geometric view is considered to consist
of curved space properties alone. [6]
As in the standard model of elementary particles, the
separation of curved space into flat space and the gravi-
tational field has the following implication: the quantum
version of RTG – if it exists – will involve quantization of
fields, but not of space.
Finally we answer the question posed in the title. The
field theoretic approach to gravitation is by itself equiva-
lent to a curved space description, so both views apply, de-
scribing the same physics from a different angle. But when
the theory is extended to the relativistic theory of grav-
itation, the bimetrism forces to describe the Minkowski
metric separately, and then we see it as most natural to
view gravitation as a field in flat space, which is Maxwell’s
view.
Topics such as a realistic equation of state for black holes
and classical tunneling of its radiation, regularization of
other singularities, as well as aspects of the inflation and
of inhomogeneous cosmology are under study.
∗ ∗ ∗
Discussion with Martin Nieuwenhuizen and Armen Al-
lahverdyan is gratefully remembered.
REFERENCES
[1] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation,
(Freeman, San Francisco, 1973).
[2] S. Weinberg, Gravitation and Cosmology: Principles and
Application of the General Theory of Relativity, (Wiley,
New York, 1972).
[3] L. D. Landau and E. M. Lifshitz, The Classical Theory of
Fields, (Pergamon, Oxford, U.K., 1951; revised 1979).
[4] S. V. Babak and L. P. Grishchuk, Phys. Rev. D61, 024038
(1999).
[5] N. Rosen, Phys. Rev. 57, 147 (1940); ibid 150; Ann. Phys.
22, 11 (1963).
[6] A.A. Logunov, The Theory of Gravity, (Nauka, Moscow,
2001).
[7] S.S. Gershtein, A.A. Logunov and M.A.Mestvirishvili, gr-
qc/0602029.
[8] S. V. Babak and L. P. Grishchuk, Int. J. Mod. Phys. D12,
1905 (2003).
[9] M. Tegmark, A. Aguirre, M.J. Rees, and F. Wilczek,
Phys. Rev D73, 023505 (2006).
[10] D. N. Spergel, et al., Astrophys. J. Suppl. 148, 175 (2003).
[11] C.M. Will, Theory and Experiment in Gravitational
Physics, (Cambridge Univ. Press, New York, 1993), chap-
ter 12.3, discusses that gravitational radiation of binary
pulsars rules out a more recent bimetric theory of Rosen,
in which black holes do not have the Schwarzschild shape.
[12] G. ’t Hooft, Nucl. Phys. B256, 727 (1985).

  Starting with a field theoretic approach in Minkowski space, the
gravitational energy momentum tensor is derived from the Einstein equations in
a straightforward manner. This allows to present them as {\it acceleration
tensor} = const. $\times$ {\it total energy momentum tensor}. For flat space
cosmology the gravitational energy is negative and cancels the material energy.
In the relativistic theory of gravitation a bimetric coupling between the
Riemann and Minkowski metrics breaks general coordinate invariance. The case of
a positive cosmological constant is considered. A singularity free version of
the Schwarzschild black hole is solved analytically. In the interior the
components of the metric tensor quickly die out, but do not change sign,
leaving the role of time as usual. For cosmology the $\Lambda$CDM model is
covered, while there appears a form of inflation at early times. Here both the
total energy and the zero point energy vanish.

arXiv:0704.0228v1  [gr-qc]  2 Apr 2007
epl draft
Einstein vs Maxwell: Is gravitation a curvature of space,
a field in flat space, or both?
Theo M. Nieuwenhuizen
Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
PACS 04.20.Cv – Fundamental problems and general formalism
PACS 04.20.Fy – Canonical formalism, Lagrangians, and variational principles
PACS 98.80.Bp – Origin and formation of the Universe
Abstract. - Starting with a field theoretic approach in Minkowski space, the gravitational energy
momentum tensor is derived from the Einstein equations in a straightforward manner. This allows
to present them as acceleration tensor = const. × total energy momentum tensor. For flat space
cosmology the gravitational energy is negative and cancels the material energy. In the relativistic
theory of gravitation a bimetric coupling between the Riemann and Minkowski metrics breaks
general coordinate invariance. The case of a positive cosmological constant is considered. A
singularity free version of the Schwarzschild black hole is solved analytically. In the interior the
components of the metric tensor quickly die out, but do not change sign, leaving the role of time
as usual. For cosmology the ΛCDM model is covered, while there appears a form of inflation at
early times. Here both the total energy and the zero point energy vanish.
It is said that in introducing the general theory of rela-
tivity (GTR), Einstein made the step that Lorentz and
Poincaré had failed to make: to go from flat space to
curved space. Technically, this arises from the group of
general coordinate transformations [1, 2]. One fundamen-
tal difficulty is then how to deal with the physics of gravi-
tation itself, since there is only a quasi energy-momentum
tensor [3]. For gravitational wave detection, e.g., this
leaves open the question as to how energy can be faithfully
transferred from the wave to the detector. The proper en-
ergy momentum tensor of gravitation was derived only
recently by Babak and Grishchuk [4], who start with a
field theoretic approach to gravitation, in terms of a ten-
sor field hµν in a Minkowski background space-time. The
metric of the latter, ηµν = diag(1,−1,−1,−1), is denoted
in arbitrary coordinates by γµν = (γ
µν)−1. The Riemann
metric tensor gµν = (g
µν)−1, is then defined by
gµν = γµν + hµν ≡ kµν , g
det(gµν)
det(γµν)
. (1)
It is just a way to code the gravitational field, allowing
to expresses distances by ds2 = gµνdx
µdxν . Such a non-
linear way to code distances in a flat space is not uncom-
mon. For diffuse light transport through clouds, one may
express distances in the optical thickness, the number of
extinction lengths. If the cloud is not homogeneous, points
at the same physical distance are described by a different
optical distance and, vice versa.
The Maxwell view that gravitation is a field in flat
space, was actually the starting point for Einstein, and
reappeared regularly. Nathan Rosen [5], coauthor of
the Einstein-Podolsky-Rosen paper that led the basis for
quantum information, considers a bimetric theory, in-
volving the Minkowski metric and the Riemann metric.
Bimetrism is quite natural, with ηµν entering e.g. particle
physics, and gµν e.g. cosmology. Rosen considers covari-
ant derivatives Dµ of Minkowski space, with Christoffel
symbols γλ
·µν vanishing in Cartesian coordinates. When re-
placing in the Riemann Christoffel symbols partial deriva-
tives by Minkowski covariant ones,
·µν =
gλσ(∂µgνσ + ∂νgµσ − ∂σgµν) 7→
·µν =
gλσ(Dµgνσ +Dνgµσ −Dσgµν), (2)
the obtained Christoffel-type symbols Gλ
·µν are tensors in
Minkowski space. Inspired by the Landau-Lifshitz and
Babak-Grishchuk results, we may define the acceleration
tensor
Aµν =
DαDβ(k
µνkαβ − kµαkνβ), (3)
where kµν = γµν + hµν and in which the γγ terms do not
http://arxiv.org/abs/0704.0228v1
Th.M. Nieuwenhuizen
contribute. Then we can calculate the combination
τµν =
Aµν − (Rµν − 1
gµνR)
. (4)
In doing so, we make use of Rosen’s observation that Rµν
remain unchanged if one replaces all partial derivatives by
covariant ones in Minkowski space [5]. It appears that
all second order derivatives drop out from (4), leaving a
bilinear form in first order covariant derivatives,
τµν =
· · :λh
· · :ρ −
· · :λh
· · :ρ
hµλ:ρhν
·λ:ρ +
kµνhλρ:σhλσ:ρ −
hλρ:µhν
hµλ:ρh · · :νλρ +
hλρ:µh · · :νλρ −
kµνhλρ:σhλρ:σ +
kµνh ·ρ:λρ h
. (5)
in which X:µ ≡ DµX and raising (lowering) of indices
· · :ρ is performed with k
µν (kµν). τ
µν is a tensor
in Minkowski space. For Cartesian coordinates, it coin-
cides with the Landau-Lifshitz quasi-tensor. In general, it
coincides with the Babak-Grishchuk tensor γtµν/g. Inclu-
sion of matter is now much easier than in [4]. Inserting
the Einstein equations in the right hand side of (4), we
may write the Einstein equations in the Newton shape:
acceleration=mass−1×force,
Aµν =
Θµν ,
Θµν =
θµν , θµν ≡ τµν + T µν . (6)
Θµν is the total energy momentum tensor of gravitation
and matter. It is conserved, DνΘ
µν = 0, since Eq. (3) im-
plies DνA
µν = 0, because covariant Minkowski derivatives
commute.
As an application, let us consider cosmology, described
by the Friedman-Lemaitre-Robertson-Walker (FLRW)
metric,
ds2 = U(t)c2dt2 − V (t)
1− kr2
+ r2dΩ2
, (7)
dΩ2 = dθ2 + sin2 θdφ2.
Let us consider flat space, k = 0, and U = 1, V (t) = a2(t)
with a the scale factor. Then ds2 = c2dt2 − a2(t)dr2
is space-independent, implying that A00 = 0, due to the
shape (3). According to (6) it then follows that the total
energy density is zero, because the gravitational energy
density, τ00 = −3c4ȧ2/(8πGa2), is negative and cancels
the one of matter, T 00 = ρ, due to the Friedman equation.
In other words, such a universe contains no overall energy.
So far we have discussed an alternative, field theoretic
formulation of GTR. If we consider a local energy mo-
mentum density as a sine qua non property, then we are
led to consider Minkowski space as a fixed “pre-space”,
that exist already without matter, just as a region of
space ahead of the earth’s orbit is right now almost empty
(Minkowskian), and when the earth arrives, there will be
more gravitational and matter fields, but, in our view, no
change of space. Also for cosmology there is a different
interpretation. In GTR coordinates are fixed to clusters
of galaxies, this is called “coordinate space”, but due to
the increasing scale factor galaxies are said to move away
from each other: physical space (i.e. Riemann space) is
said to expand. Here we are led to another view: Coordi-
nate space is physical space, so clusters of galaxies do not
move away from each other in time. [6] However, the cos-
mic speed of light dr/dt = c/a(t), which was very large at
early times, keeps on decreasing, thus causing a redshift,
till a is infinite, when galaxies are invisible.
Relativistic Theory of Gravitation, RTG. Let us move
on to an extension of GTR, giving up general coordinate
invariance. Discarding a total derivative of the Hilbert-
Einstein action, Rosen expresses the gravitational action
d3xdt
−g LR in terms of [5]
c4gµν
·λσ −Gλ·µσGσ·νλ) =
128πG
× (2hµν:ρhµν:ρ − 4hµν:ρhµρ:ν − hν·ν:µh ·ρ:µρ ).
Involving only Minkowski covariant first order derivatives,
it is close to general approaches in field theory. Logunov
and coworkers continue on this [6]. The subgroup of gauge
transformations that transform hµν but leave coordinates
invariant, allows three extra terms [6],
Lg = LR − ρΛ +
ρbiγµνg
µν − ρ0
γ/g. (9)
Here ρΛ is the familiar energy related to a cosmological
constant. The ρ0 term describes a harmless shift of the
zero level of energy, δS = −
d3xdt
−γρ0. The bimetric
term ρbi couples the Minkowski and the Riemann metrics.
It acts like a mass term, because it breaks general coor-
dinate invariance, and has some analogy to a mass term
in massive electrodynamics. Logunov then imposes the
relation
ρΛ = ρbi = ρ0, (10)
which, in the absence of matter, keeps space flat, hµν = 0,
gµν = γµν and also Lg = 0. Thus one free parameter re-
mains. Logunov’s choice ρbi ≡ −m2c4/(16πG) < 0 leads
to an inverse length m and, in quantum language, a gravi-
ton mass h̄m/c. The negative cosmological constant can
be counteracted by an inflaton field [7]. The obtained
theory has some drawbacks, such as self-repulsive prop-
erties for matter falling onto a black hole, and a minimal
and a maximal size of the scale factor in cosmology [6] [7].
For a related approach to finite range gravity, based on a
generalized Fierz-Pauli coupling, see [8].
Einstein vs Maxwell
We shall focus on the opposite choice, a positive cosmo-
logical constant Λ, [9] [10]
Ωv,0H
9.78Gyr
ρbi ≡
= ρΛ. (11)
Now the graviton has an “imaginary mass”, m =
−2Λbi/c, it is a “tachyon”: Gravitational waves are
unstable at today’s Hubble scale. But this is of no con-
cern, since on that scale, not single gravitational waves
but the whole Universe matters, being unstable (expand-
ing) anyhow.
Though we take ρbi = ρΛ, Λbi = Λ, our further notation
is valid for the general case ρbi 6= ρΛ, Λbi 6= Λ.
The Einstein equations that couple the Riemann metric
to matter read
Rµν −
gµνR =
tot , (12)
tot = T
µν + ρΛg
µν + ρbiγρσ(g
µρgσν − 1
gµνgρσ).
Conservation of energy momentum, T
tot;ν = 0, imposes a
constraint due to the ρbi terms, [6]
= 0, or Dνh
µν = 0, (13)
which for Cartesian coordinates coincides with the GTR
harmonic condition ∂ν(
−ggµν) = 0 [2]. Thus the the-
ory automatically demands the harmonic constraint for
gµν , or, equivalently, the Lorentz gauge for hµν , thereby
severely reducing the gauge invariance of GTR.
Changes of Einstein’s GTR have mostly met deep trou-
bles with one or another established property, though not
all proposals are ruled out [1,11]. The present one is rather
subtle and promising. For most applications, the Hubble-
size ρΛ = ρbi terms in Eq. (11,12) are too small to be
relevant, so known results from general relativity can be
reproduced. Indeed, viewed from a GTR standpoint, Eq.
(13) is only a particular gauge, and actually often con-
sidered, while the cosmological constant only plays a role
in cosmology. Logunov checked a number of effects in
the solar system: deflection of light rays by the sun, the
delay of a radio signal, the shift of Mercury’s perihelion,
the precession of a gyroscope, and the gravitational shift
of spectral lines. [6] Likewise, we expect agreement for
binary pulsars. [11] Differences between GTR and RTG
may arise, though, for large gravitational fields, that we
consider now.
Black holes. It is known that true black holes, ob-
jects that have a horizon, do not occur in the RTG with
ρΛ, ρbi → 0. [5] But there are solutions very similar to it,
that might be named “grey holes”, but we just call them
“black holes”. The Minkowski line element in spherical
coordinates is simply γµνdx
µdxν = c2dt2 − dr2 − r2dΩ2.
The one of Riemann space is
ds2 = gµνdx
µdxν = U(r)c2dt2 − V (r)dr2 −W 2(r)dΩ2. (14)
In harmonic coordinates, the Schwarzschild black hole is
described by [2]
r − rh
r + rh
, Ws = r + rh, rh =
. (15)
The horizon radius rh equals half the Schwarzschild radius.
Let us scale r → rrh, and define
U = eu, V = ev, W = 2rhe
w, (16)
so that w is small near the horizon. The dimensionless
small parameter arising from ρbi = ρΛ, is very small,
λ̄ ≡ rh
2Λ = 2.38 10−23
. µ̄ ≡ rh
2Λbi = λ̄. (17)
The sum and difference of the (t, t) and (r, r) Einstein
equations give
ev−2w − w′(u′ − v′ + 4w′)− 2w′′
= ev(λ̄2 − 1
µ̄2r2e−2w) +
8πGr2h
ev(ρ− p), (18)
w′(u′ + v′ − 2w′)− 2w′′
µ̄2(ev−u − 1) + 8πGr
ev(ρ+ p), (19)
respectively. The harmonic condition imposes
u′ − v′ + 4w′ = r exp(v − 2w).
In the Schwarzschild black hole of GTR, there is no
matter outside the origin. We shall focus on that situation.
A parametric solution of these equations then reads
1 + η(eξ + ξ + log η + r0)
1− η(eξ + ξ + log η + r0)
, (20)
u = ξ + log η,
v = ξ − ln η − 2 log(eξ + 1), (21)
w = ηeξ + µ̄2(ξ + log η + w0).
where ξ is the running variable and η is a small scale.
Corrections of next order in η can be expressed in diloga-
rithms, but they are not needed since µ̄ is very small.
To fix the scale η, we note that energy momentum con-
servation implies, as in GTR, (ρ + p)u′ + 2p′ = 0. In
the stationary state all matter is located at the origin,
which is only possible if p(r) ≡ 0, implying ρ(r)u′(r) = 0.
This is obeyed for r 6= 0 since ρ = 0 there, but since
ρ(0) > 0 (it is infinite), we have to demand u′(0) = 0.
Let us define a factor α by α = µ̄2/η. The above solu-
tion brings w′(r) = ∂ξw/∂ξr = (e
ξ + α)/[2(eξ + 1)], so in
the interior w′ = 1
α. Since ev ≪ 1 there, Eqs. (18,19)
confirm that w′′ = 0, and with w(1) = O(η) this solves
Th.M. Nieuwenhuizen
1.510.5
Fig. 1: Black hole functions U(r), V (r) and W (r), scaled
by factors 10, (bold lines) for µ̄ = 0.1, compared to the
Schwarzschild solution (thin lines; the part V < 0 for r < 1 is
not shown). Inside the horizon, U and V decay very rapidly.
Since they remain positive, time keeps its role in the interior.
w(r) = 1
α (r − 1). Moreover, from the harmonic con-
straint (20) we have in the interior u(r)− v(r) + 4w(r) =
const = 2 ln η, implying that Eq. (19) yields in the interior
u′(r) = {exp[2α(r − 1)]− η2 − µ̄2}/(2η). From u′(0) = 0
we can now solve α,
α = log
, η =
ln 1/µ̄
. (22)
As seen in fig. 1, our solution (16,20,-22) coincides with
Schwarzschild’s for ξ ≫ 1. In the regime ξ = O(1), there
is a transition towards the interior ξ ≪ −1, where ex-
ponential corrections can be neglected. Both U = ηeξ
and V = eξ/η are very small there, but, contrary to the
Schwarzschild case, they remain positive: The behavior in
the interior of the RTG black hole is not qualitatively dif-
ferent from usual, be it that the gravitational field is large.
Width of the brick wall. The transition layer ξ = O(1)
acts like ’t Hooft’s brick wall, [12] of characteristic width
ℓ⋆ = ηrh. Comparing to the Planck length ℓP =
h̄G/c3,
we get
0.977 10−9
1 + 0.019 log(M/M⊙)
. (23)
If quantum physics sets in at the Planck scale, our ap-
proach makes sense only for M > 103 M⊙.
Motion of test particles. For RTG with a negative cos-
mological constant, [6] it was claimed that an incoming
spherical shell of matter is scattered off from a black hole,
a counter-intuitive finding. Let us reconsider this issue.
The motion of a test body occurs along a geodesic
+ Γµνρv
νvρ = 0, vµ =
. (24)
For spherical shells of in-falling matter one needs Γ001 =
U ′/(2U). This brings dt/ds = v0 = 1/(CiU), for some
Ci. Solving v
1 = dr/ds from gµνv
µvν = 1, we then get
dr/dt = (ds/dt)(dr/ds) = −c
U(1− C2i U)/V . We can
now fix Ci at the initial position r = ri, where the spherical
shell is assumed to have a speed dri/dt = vi = βic
Ui/Vi,
viz. Ci =
(1 − β2i )/Ui, with |βi| ≤ 1. The differential
proper time dτ =
Udt and length dℓ =
V dr bring in
the particle’s rest frame dℓ/dτ =
V/Udr/dt, yielding
1− U(r(τ))
U(ri)
(1 − β2i ). (25)
The extreme case is when βi = 0 at ri = ∞, dℓ/dτ =
1− U. To have |dℓ/dτ | < c, it thus suffices that 0 <
U ≤ 1, which is the case. Near the horizon, |dℓ/dτ | is
almost equal to c and the more the shell penetrates the
interior, the closer its speed gets to c. For an outside
observer, the time to see it hit the center of the hole,
dr/|ṙ| equals (rh/c)
dr exp[ 1
(v − u)]. It is finite
and predominantly comes from the horizon, T = rh/cµ̄
2.74× 1032M/M⊙ yr.
The approaches [6–8] have a similar a black hole. While
[8] properly has U ′(0) = 0, in Logunov’s case one has
µ̄2 < 0, so w′ = 1
α < 0 in the interior. This seems to
solve the paradox of “matter reflected by the black hole”:
In-falling matter just enters, but the Logunov coordinate
x = exp(w) − 1 is non-monotonic (x′ < 0 in the interior).
However, the situation is more severe: For α < 0, the
theory does not allow a solution with u′(0) = 0, depriving
that theory of a proper black hole. This condition can
neither be obeyed in GTR: If the central mass is slightly
smeared, the Schwarzschild black hole cannot obey energy-
momentum conservation in GTR.
Cosmology. Starting from the FLRW metric, the har-
monic condition brings two relations: U ∼ V 3 and k = 0:
Minkowski space filled homogeneously with matter re-
mains flat [6]. We may thus put U = a6(t)/a4
, V = a2(t).
Going from cosmic time t to conformal time τ =
a3a−2
yields the familiar Einstein equations, extended by Λbi
terms,
− Λbi
, (26)
= −4πG
(ρ+ 3p) +
− Λbi
The first is the modified Friedman equation, the second
corresponds to the first law d(ρtota
3) = −ptotda3 provided
we define ρtot = ρ+ρΛ+ρ2+ρ6 and ptot = p−ρΛ− 13ρ2+ρ6,
with ρ2 = −3ρbi/2a2 and ρ6 = ρbia4∗/2a6. Note that ρ2
acts as a positive curvature term.
The scale factor has an absolute meaning. If we as-
sume that a ≫ 1 and a ≫ a2/3∗ , Eq. (26) just coin-
cides with the ΛCDM model (cosmological constant plus
cold dark matter), that gives the best fit of the observa-
tions [9] [10]. The ρ2 term allows a positive curvature-
type contribution. At large times, there is the exponential
growth a(τ) = C exp(H∞τ) with H∞ = c
Λ/3. In cos-
mic time this reads a(t) = a
∗ [3H∞(t0 − t)]−1/3, where
Einstein vs Maxwell
t0 is “the end of time”, the moment where the scale factor
has become infinite. The minimal scale factor is zero: in
this theory a big bang can occur since ρbi > 0. With-
out including an inflaton field, Eq. (26) yields an initial
growth of the expansion a = (a2
3Λbi/2)
1/3. In cos-
mic time this reads a = a1 exp(ct
Λbi/6), i. e., a certain
inflation scenario starting at t = −∞.
Also in RTG the gravitational energy precisely compen-
sates the other energy contributions at all times. The
vacuum energy also vanishes: In empty space, the cosmo-
logical constant energy ρΛ cancels the ρbi terms, due to
Eq. (10). See Eq. (12) for gµν = γµν .
In conclusion, we have first written the Einstein equa-
tion in a form that involves the gravitational energy
momentum tensor. An underlying Minkowski space is
needed, in which gravitation is a field. The metric ten-
sor is a way to deal with it, but the equations for the field
itself exist too, see Eq. (6). For flat cosmology it follows
that the total energy vanishes.
Next we have broken general coordinate invariance by
going to the bimetric theory of Logunov, called Relativis-
tic Theory of Gravitation. We have shown that the choice
of a positive bimetric constant allows to regularize the
interior of the Schwarzschild black hole: time keeps its
standard role and escape is, in principle, possible. While
neither the Schwarzschild nor the Logunov black hole sur-
vives smearing of the central mass by a tiny pressure in the
equation of state, ours does. Our modification of the Ein-
stein equations involves the cosmological constant, so it is
of Hubble size, immaterial for solar problems. In cosmol-
ogy, the theory directly leads to the ΛCDM model, while
it could accommodate a positive curvature-like term. At
short times, there is a form of inflation. The gravitational
energy exactly compensates the material energy. The zero
point energy vanishes (“again”), though the cosmological
constant is finite and positive: It is canceled by the bimet-
ric terms.
Euclidean space, a special case of Riemann geometry,
seems to be invoked by Nature, at least far away from bod-
ies and in cosmology. Our approach supports the follow-
ing space-time interpretation: curvature is a geometric de-
scription of the gravitational field in flat space. Clusters of
galaxies do not move away from each other, but the speed
of light changes with cosmic time, dr/dt = [a(t)/a∗]
while the conformal speed is dr/dτ = c/a(τ) as usual.
An empirical way to establish the Minkowski metric is
to present the Einstein equations as (c4/8πG)Rµν −Tµν +
gµνT + ρΛgµν = ρbiγµν , and to measure the left hand
side, which in the geometric view is considered to consist
of curved space properties alone. [6]
As in the standard model of elementary particles, the
separation of curved space into flat space and the gravi-
tational field has the following implication: the quantum
version of RTG – if it exists – will involve quantization of
fields, but not of space.
Finally we answer the question posed in the title. The
field theoretic approach to gravitation is by itself equiva-
lent to a curved space description, so both views apply, de-
scribing the same physics from a different angle. But when
the theory is extended to the relativistic theory of grav-
itation, the bimetrism forces to describe the Minkowski
metric separately, and then we see it as most natural to
view gravitation as a field in flat space, which is Maxwell’s
view.
Topics such as a realistic equation of state for black holes
and classical tunneling of its radiation, regularization of
other singularities, as well as aspects of the inflation and
of inhomogeneous cosmology are under study.
∗ ∗ ∗
Discussion with Martin Nieuwenhuizen and Armen Al-
lahverdyan is gratefully remembered.
REFERENCES
[1] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation,
(Freeman, San Francisco, 1973).
[2] S. Weinberg, Gravitation and Cosmology: Principles and
Application of the General Theory of Relativity, (Wiley,
New York, 1972).
[3] L. D. Landau and E. M. Lifshitz, The Classical Theory of
Fields, (Pergamon, Oxford, U.K., 1951; revised 1979).
[4] S. V. Babak and L. P. Grishchuk, Phys. Rev. D61, 024038
(1999).
[5] N. Rosen, Phys. Rev. 57, 147 (1940); ibid 150; Ann. Phys.
22, 11 (1963).
[6] A.A. Logunov, The Theory of Gravity, (Nauka, Moscow,
2001).
[7] S.S. Gershtein, A.A. Logunov and M.A.Mestvirishvili, gr-
qc/0602029.
[8] S. V. Babak and L. P. Grishchuk, Int. J. Mod. Phys. D12,
1905 (2003).
[9] M. Tegmark, A. Aguirre, M.J. Rees, and F. Wilczek,
Phys. Rev D73, 023505 (2006).
[10] D. N. Spergel, et al., Astrophys. J. Suppl. 148, 175 (2003).
[11] C.M. Will, Theory and Experiment in Gravitational
Physics, (Cambridge Univ. Press, New York, 1993), chap-
ter 12.3, discusses that gravitational radiation of binary
pulsars rules out a more recent bimetric theory of Rosen,
in which black holes do not have the Schwarzschild shape.
[12] G. ’t Hooft, Nucl. Phys. B256, 727 (1985).
