The Question of Validity of the ``Dynamic Solutions'' Constructed by Christodoulou and Klainerman

Abstract

Christodoulou and Klainerman claimed the existence of "dynamic" solutions for the vacuum Einstein equation of 1915. However, their so-called dynamic solutions are merely constructed from their presumed strong asymptotically flat (SAP) "initial data sets" without showing the physical relevance. They have not shown the existence of a case other than the static solutions. Moreover, they have not related any of their dynamic solutions to dynamic sources. Thus, their claim of the existence of a dynamic solution is invalid simply because it has not been proven. It is shown that for a presumed dynamic solution, an SAF initial data set is incompatible with the Maxwell-Newton approximation, the linear field equation for weak gravity, which is due to the equivalence principle. It is proven also that a dynamic SAF initial data set is inconsistent with Einstein's radiation formula, which is supported by experiments. It is concluded that the only physically valid SAF initial data sets are the static solutions.

Full text

The Qu es tion of Va lidi ty of th e
“ Dy na mic Solu tions ” Constr uc te d
by Christ o doulo u an d Klain e r ma n
C. Y. Lo Applied and Pure Research Institute, 17 Newcastle Dr., Nashua, NH 03060, USA
Abstract
Christodoulou and Klainerman claimed the existence of “dynamic” solutions for the
vacuum Einstein equation of 1915. However, their so-called “dynamic” solutions
are merely constructed from their presumed strong asymptotically flat (S.A.F.)
“initial data sets” without showing the physical relevance. They have not shown the
existence of a case other than the static solutions. Moreover, they have not related
any of their “dynamic” solutions with dynamic sources. Thus, their claim of the
existence of dynamic solution is invalid because simply it has not been proven. It is
shown that for a presumed dynamic solution, a S.A.F. initial data set is incompatible
with Maxwell-Newton approximation, the linear field equation for weak gravity,
which is due to the equivalence principle. It is proven also that a “dynamic” strong
asymptotically flat initial data set is inconsistent with Einstein’s radiation formula,
which is supported by experiments. It is concluded that the only physically valid
S.A.F. initial data sets are the static solutions.
Key words: Einstein’s radiation formula, dynamic solution, dynamic source,
Maxwell-Newton approximation, causality, and boundedness
1. INTRODUCTION
A major problem in general relativity, as pointed out by Kramer, Stephani, Herlt & MacCallum [1], is that any
Riemannian geometry metric with the proper metric signature would be accepted as a valid solution of Einstein equation
of 1915. Consequently, as pointed out by Kinnersley [2], many unphysical solutions are accepted as valid. Thus, a crucial
problem for a mathematical solution is its validity in physics.
The nonlinear nature of the 1915 Einstein equation certainly gives surprises. In particular, it has never been verified
that the Einstein equation of 1915 has bounded dynamic solutions. In 1936 Einstein himself discovered [3] the non-
existence of any propagating gravitational wave. But, “plane-waves” were proposed by Bondi, Pirani and Robinson [4] in
1959. Subsequently, time-dependent solutions were found for the gravity of electromagnetic plane-waves [5,6]. In 1991, it
was found, however, that such solutions are incompatible with the physical requirements such as Einstein’s notion of
weak gravity, the principle of causality and the equivalence principle [7-9]. Moreover, it has been shown [8] that there are
no bounded gravitational plane-waves. The recent wave solution proposed by Au, Fang and To [10] is also found to be
unbounded. In short, all known wave solutions in vacuum are unbounded and unphysical.
In 1953, Hogarth [11] conjectured that a dynamic solution for the Einstein equation does not exist. (The existence of
dynamic solution was questioned implicitly by Gullstrand [12,13] in 1921.) It is known [14] that any attempt to extend the

Maxwell-Newton approximation to higher approximations leads to divergent terms. In 1993, it has been proven [15] that
for a dynamic problem of particles the linear equation of weak gravity, as a first order approximation, is incompatible
with the nonlinear Einstein field equation. Moreover, the Einstein equation does not have a physical dynamic solution for
weak gravity unless the gravitational energy tensor with an anti-gravity coupling is added to the source (see also Section
3). The necessity of an anti-gravity coupling term manifests why a bounded wave solution is impossible for the 1915
version.
Nevertheless, Christodoulou and Klainerman [16] claimed their constructions on “dynamic” solutions for Einstein
equation of 1915. But, in 1996 a book review published in ZFM (see Appendix A), pointed out that the authors have made
some unexpected mistakes, and the mathematical proof of the main theorem is difficult to follow, and suggested the main
conclusion of this book may be unreliable. However, to many readers, a suggestion of going through more than 500 pages
of mathematics is not a very practical proposal.
Since this book supports the faith on the existence of dynamical solutions, in view of the lack of other supports, this
book is often cited without referring to the mathematical details. In view of the fact that unphysical solutions were often
accepted as valid [1,2,5-8,15], one should examine first whether their claims in physics that matters are supported by their
claims in mathematics. In this paper, it is shown that the claimed dynamic nature of their solutions is actually implicitly
assumed, but not proven [9].
In physics, a dynamic solution must be related to dynamic sources. A “time-dependent” solution in vacuum may not
necessarily be a physical solution (Section 2). To begin with, they construct their solutions based on mathematics whose
presumed physical validity are dubious. For instance, their “initial data sets” can be incompatible with the field equation
for weak gravity [17,18] (Sections 4 & 5). Second, the only known cases are static solutions, and no time-dependent
solution has been obtained or proven to illustrate the claimed dynamic nature. Third, they have not been able to relate any
of their constructed solutions to a dynamic source. (Mathematically, it is known that a source would put severe constraints
on a possible solution.)
In pure mathematics, it is well known that if no example can be given, such abstract mathematics is likely wrong [19].
The claim of Christodoulou and Klainerman is questionable since the existence of their dynamic solutions is presumed but
not proven. Here, it will be shown that their basic assumption, the existence of the strong asymptotically flat initial data
sets has no dynamic content in physics; otherwise it would be inconsistent with Einstein’s radiation formula as well as the
field equation for weak gravity (Sections 3-5). This supports the book review that the mathematics would be unreliable
(see Appendix A).
In fact, the claim of Christodoulou and Klainerman is also mathematically incorrect because not only is there no proof
for the existence of a bounded dynamic initial condition, but also the incompatibility of their S.A.F. condition for a
dynamic situation (see Section 4) with the linear field equation for weak gravity (Section 3) means that such a dynamic
condition is impossible for the Einstein equation of 1915.
2. RADIATIVE DYNAMIC SOLUTIONS AND CLAIMS OF CHRISTODOULOU AND KLAINERMAN
Some theorists believe that related to a field equation, there are separate types of question as follows: 1) the validity of
deriving the field equation in physics; 2) restrictions on solutions of these equations which may be required for their
physical relevance; 3) the existence of solutions of the equations with certain properties. For example, because the existing
basic laws of electricity and magnetism were inconsistent, the Maxwell equation was derived by considering the continuity
equation. The solutions of Maxwell equation are restricted by boundary conditions. Moreover, a new type of solutions,

wave solutions, was discovered. However, a new type of solution need not have meaning in physics unless proven
otherwise. For instance, the wave solution is generally accepted only after Hertz demonstrated the existence of the
electromagnetic wave.
A major problem in the current theory of relativity is that there is confusion between mathematical solution and
physical solutions [1,2]. A mathematical solution of a differential equation requires only a satisfaction of the equation. A
physical solution is a mathematical solution for a given physical situation, and therefore must satisfy the related physical
requirements. The problem is that physical requirements are often ignored.
A wave solution in vacuum, by definition, is a physical solution. Waves, which transfer energy, must be related
dynamic sources, according to the principle of causality [9]. However, for some situations, there are no formal source, a
plane-wave is a spatial local idealization of a weak wave from a distant source [4,8,9]. To emphasize that fact that a wave
is related to the dynamic of its source, a solution, which contains wave components, is called a dynamic solution. Thus, to
claim a “time-dependent” solution as a dynamic or radiative solution, one must relate such a solution to a dynamic source.
Note that a “time-dependent” solution in vacuum may not be a valid solution in physics, in particular, if such a
solution is unrelated to a dynamic source. This is well known in electrodynamics. The amplitude of an electromagnetic
radiation has approximately the form of A(t-r)/r (where A(r) is an arbitrary but bounded function of the radial r). But,
there are multi-pole fields, which have an approximate form whose radical dependence is 1/r l+1 (l ≥ 1). Although such
multiple fields are solutions in vacuum, any of them alone may not be related to a dynamic source. Moreover, some time-
dependent solutions in vacuum are not related to the radiation of an isolated system because no dynamic source can be
related to such a solution. For instance, the function A0exp (t - z)2 (where A0 is a constant) cannot be related to an
isolated dynamic source.
For the Einstein equation, the situation is even more complicated due to the following:
1) Unlike the Maxwell equation that is due to necessity of consistency, Einstein equation of 1915 is the guesswork of a
genius, Albert Einstein. As Klien [20] pointed out, there is no satisfactory proof of rigorous validity for the Einstein
equation. Experiments support Einstein equation only for the static case [15]. Thus, there is no certainty to expect
that Einstein equation has a solution for a dynamic situation.
2) General relativity is based on the equivalence principle and the general principle of relativity which states [21] that
“if it is necessary for the purpose of describing nature, to make use of a coordinate system arbitrarily introduced by us,
then the choice of its state of motion ought to be subject to no restriction.” But, general mathematical covariance can
be incompatible with the equivalence principle [22,23]. Thus, the current gauge notion (rejected by Eddington [24]
and questioned implicitly by Einstein [25]) actually must be restricted by the satisfaction of the equivalence principle
[23].
3) Therefore, the incompatibility between the Einstein equation and its linearization need not mean that the linear field
equation is problematic. The problem can be that both the Einstein equation and the mathematical derivation for the
linear equation are not valid for a dynamic situation. In particular, the linear field equation, as a first order
approximation, is supported by all experiments [15,26]. Indeed, it has been found that the binary pulsar radiation
experiments cannot be explained without modifying the 1915 version [15]. Moreover, as a first order approximation,

not only is the linear equation supported by experiments, but also it can be derived from physical principles, which
lead to general relativity [27].
4) It is well known that a “time-dependent” solution in vacuum may not be valid in physics. There is no reason to expect
that the Einstein equation is an exception. In view of the fact that Einstein equation has not been proven to be valid
for a dynamic case, claiming the existence of dynamic solutions based on their time-dependence alone, is clearly
inadequate.
However, it seems, these problems were not recognized by Christodoulou and Klainerman [16]. In particular, they have
made no effort to relate dynamic sources to their solutions.
Nevertheless they claimed, in their own words, that their work accomplished the following 4 goals:
1. It provides a constructive proof of global, smooth, nontrivial solutions to the Einstein-Vacuum equations, which look,
in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities.
2. It provides a detailed description of the sense in which these solutions are close to the Minkowski space-time in all
directions and gives a rigorous derivation of the laws of gravitational radiation proposed by Bondi. It also describes
our new results concerning the behavior of the gravitational field at null infinity.
3. It obtains these solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the
initial data set of the Minkowski space-time, and thus it establishes the global dynamic stability of the latter.
4. Though our results are established only for developments of initial data sets which are uniformly close to the trivial
one, they are in fact valid in the complement of the domain of influence of a sufficiently large compact subset of the
initial manifold of any “strongly asymptotically flat” initial data set.
It should be noted that the claim of “dynamic developments” by Christodoulou and Klainerman means only time-
dependency. Moreover, they have not shown the existence of a bounded solution with time-dependency.
Thus, their “dynamic developments” actually may not have any dynamic content since they are not related to any
dynamic source. In general relativity, as pointed out by Low [28], a time-dependency can be achieved by simply having a
coordinate transformation. Since their solutions have not been related to a dynamic source, it is not clear whether their
solutions are valid in physics. Since they have not provided any concrete time-dependent example which can illustrate
their claimed validity as dynamic solutions, it is possible that their physical initial data sets consist of only static solutions
(see also Section 4).
This work of Christodoulou and Klainerman is actually more in the area of pure mathematics, although they use the
terminology of physicists, state their goals in physics, and implicitly accept physical requirements such as compatibility
with Einstein’s notion of weak gravity. But, they ignore existing theory of gravitational radiation [17,18] and the principle
of causality which requires relating their solutions to dynamic sources.
3. THE EINSTEIN EQUATION OF 1915 AND THE LINEAR FIELD EQUATION FOR WEAK GRAVITY
To examine the physical validity of the solutions of Christodoulou and Klainerman, it is necessary to analyze the
validity of Einstein equation for weak gravity. It has been shown that the linear field equation can be derived from the
equivalence principle and related physical considerations, and that the 1915 Einstein equation is the consequence of an
over simplification which can be valid only for the static case [15,27].
For the convenience of analysis, let us review first the relationship between the Einstein equation and the linear field
equation. The non-linear Einstein field equation of 1915 [25] is

Gab ≡ Rab −
1
2
gabR = - K T(m) ab (1a)
where its source T(m)ab is the energy-stress tensor for massive matter and depends on the space-time metric gab; Rab is
the Ricci curvature tensor. Note that for Gat (a = x, y, z, t), there is no second order time derivatives [18,29]. Thus, there
are four equations to restrict the initial condition of a Cauchy problem.
The harmonic gauge condition, whose application was questioned by Einstein [25], is
∂
∂
xa
(g
1
2
gab) = 0 (1b)
where g is the determinant of the metric gab. Note that Eddington [24] rejected this gauge and the belief that any Gauss
system is a valid space-time coordinate system. In fact, a Gauss space-time coordinate system can be incompatible with
the equivalence principle [23,27]. Physics requires that for a weak source, as shown in the static case of eq. (1a), a weak
gravity should be produced. Mathematically, however, eq. (1a) may not have a weak dynamic solution since nothing
guarantees the general validity of eq. (1a).
Assuming the deviation from the flat metric is small, linearized gravity is a result of neglecting all terms of explicit
second order of deviations. Consequently, eq. (1a) and eq. (1b) are respectively linearized to be
G(1)ab = -KT(m)ab, where G(1)ab ≡ Gab - G(2)ab =
1
2
∂c∂ c
γ
ab + H(1)ab (2a)
where H(1)ab ≡ -
1
2
∂ c[∂ b
γ
ac + ∂a
γ
bc] +
1
2
ηab∂ c∂ d
γ
cd ;
and
∂ c
γ
cb = 0 , (2b)
where
γ
ab = γab -
1
2
ηab , γab = gab - ηab , γ = ηcd γcd .
The gauge (2b) implies that H(1)ab = 0 and that eq. (2a) is reduced to the linearized field equation,
1
2
∂c∂ c
γ
ab = -KT(m)ab , (3a)
and
γ
ab(xi ,t) = −
K
2
π
∫
1
R
Tab [yi, (t - R)] d3y, where R2 =
( )
x y
i i
i
−
=
∑
2
1
3
(3b)
Solution (3b) shows that eq. (3a) satisfied the physical requirement, compatibility with the notion of weak gravity.
However, this may not be satisfied by eq. (1a). Unlike eq. (1a), the Cauchy initial condition of eq. (3a) can be arbitrary.
This suggests that, for a dynamic case, eq. (3a) and eq. (1a) can be incompatible.
Thus, in linearized gravity, there are two independent equations namely: the linear field equation (3a) and the
linearized gauge condition (2b). It should be noted that, however, these equations together require the linearized
conservation law, ∂ aTab = 0, which implies no radiation [30,31]. But, instead one can use

∇aTab = ∂ aTab + ΓacaTcb - ΓcbaTac = 0, (4)
the conservation law. Thus, if eq. (2b) is not needed to obtain eq. (3a), the derivation of the radiation formula could
become valid. However, since eq. (2a) directly implies linearized conservation law, there is little hope that Einstein’s
radiation formula can be justified with linearized gravity.
On the other hand, since eqs. (3a) and (4) imply H(1)ab is of the second order [15,24], one might hope that eq. (3a) may
still give the first order approximation of eq. (1a) even though (2b) is not exact. But, this is impossible since eq. (3a) has
been proven to be not a dynamic approximation of eq. (1a) [15]. Therefore, it is more appropriate to call eq. (3a) as
Maxwell-Newton Approximation.
Nevertheless, some argued that the wave component in gat (for a = x, y, z, or t) as artificially induced by the harmonic
gauge [29]. This is resolved by that eq. (3a) can be justified independently by physical principles which lead to general
relativity. The equivalence principle implies that the geodesic equation,
d x k
ds
kab
dx a
ds
dxb
ds
2
20
+ =Γ
, where
( )
Γ
kab gkn
agnb b gna n gab
= + −
1
2
∂ ∂ ∂
(5)
is the equation of motion for a neutral particle. In comparison with the gravitational potential Φ in Newton’s theory, one
obtains c2gtt/2 ≈ Φ which satisfies the Poisson equation ∆Φ = 4πκρ , where ρ is the mass density and κ is the coupling
constant. Then, one has the field equation, ∆gtt/2 ≈ 4πκc-2Ttt, where Ttt ≈ ρ. Then, according to special relativity and the
Lorentz invariance, one has the equation for the first order,
1
2
∂c∂ cgab =
1
2
∂c∂ cγab = −4πκc-2[αT(m)ab + βZ(m)ηab], (6a)
where α and β are constants which satisfy α + β = 1, and Z(m) = T(m)cdηcd.
To have the exact equation, since the left hand side of eq. (6a) does not satisfy the covariance principle, one must
search for a tensor whose difference from
1
2
∂c∂ cγab is of second order in κc-2. In Riemannian geometry, it has been
proven [18] that the curvature tensor Rλµνκ is the only tensor that can be constructed from the metric tensor and its first
and second derivatives, and is linear in the second derivatives.” Thus, Einstein identified the Ricci curvature tensor Rab
(≡ Rλaλb) as the required tensor; and if Rab includes no net first order term other than
1
2
∂c∂ cγab , the exact field
equation would be
Rab = X(2)ab − 4πκc-2[αT(m)ab + β T(m)gab], (7a)
where T(m) = gcdT(m)cd and X(2)ab is a second order unknown tensor chosen by Einstein to be zero. However, a non-
zero X(2)ab may be needed to ensure eq. (6a) as an approximation of eq. (7a) [15].

According to eq. (6a), eq. (4), and the requirement that there is no other net first order term in R ab (i.e., the sum ∂
c[∂bγac + ∂a γbc ] - ∂a∂bγ is of second order), it is necessary to have α = 2, and β = -1. Thus, we have an equation for the
first order approximation, which is equivalent to eq. (3a),
1
2
∂c∂ cγab) = −8πκc-2[T(m)ab +
1
2
Z (m)ηab] . (6b)
This field equation of massive matter has been determined by physics. The solution of eq. (6) is compatible with the
equivalence principle as demonstrated in Einstein’s [25] calculation of the bending of light. Note, however, the validity of
eq. (3a) is limited to the case when the source term is a massive energy tensor.
Note that Einstein obtained the same values for α and β by considering eq. (7a), after assuming X(2)ab = 0 [32]. An
advantage of the approach of considering eq. (6) is that the assumption X (2)ab = 0 is not needed. Then, it is possible to
obtain from eq. (7a) an equation different from eq. (1a),
Gab ≡ Rab −
1
2
gabR = - K [T(m)ab − Y(2)ab], (7b)
where
K = 8πκc-2 , and KY(2)ab = X(2)ab -
1
2
gab{X(2)cdgcd}
is of second order. The conservation law (4), ∇cT(m)cb = 0 and ∇cGcb ≡ 0 implies also
∇aY(2)ab = 0. (7c)
Y(2)ab has been proven as the gravitational energy tensor of t(g)ab whose exact form has not been found [15]. Note also
that eq. (3a) is an approximation of eq. (7), but needs not be an approximation of (1a).
The anti-gravity coupling of t(g)ab is due to Einstein’s radiation formula and it explains the failure of eq. (1a) [15].
The possibility of such a coupling was first pointed out by Pauli [33]. The need of an anti-gravity coupling was first
discovered in calculating the gravity of an electromagnetic wave [7,8].
In short, based on the equivalence principle, an appropriate field equation of first order approximation for massive
matter is eq. (3a). On the other hand, eq. (1a) is shown to be a result of an over simplification that the gravitational
energy-stress tensor is assumed to be zero. This explains why eq. (1a) must be modified for the binary pulsar experiments.
In contrast to intuition, Newtonian theory is not a dynamic limit of eq. (1a). Eq. (3a) is called Maxwell-Newton
Approximation [15] and it is on the most solid theoretical ground possible within general relativity. Thus, it is difficult to
question “whether solutions derived by the usual method of linear approximation necessarily correspond in every case to
exact solutions [34]”.
For a dynamic situation, eq. (3a) and eq. (1a) are incompatible because it has no bounded dynamic solution. From this
analysis it should be note however, that eq. (3a) should have been a valid approximation related to any bounded solution
of eq. (1a) for weak gravity. Thus, the S.A.F. initial condition of Christodoulou and Klainerman, if it is related to a
dynamic source, should have been compatible with eq. (3a).

4. THE QUESTIONABLE DYNAMICS OF STRONG ASYMPTOTICALLY FLAT INITIAL DATA SETS
Christodoulou and Klainerman [16] consider a 3+1-dimensional manifold M and an Einstein metric g. There exists a
Cauchy hypersurface in M梐 hypersurface ä with the property that any causal curve intersects it at precisely one point.
They allow the existence of a globally defined differentiable function t whose gradient Dt is timelike everywhere. They
call t a time function, and the foliation given by its level surfaces a t-foliation. Topologically, a space-time foliated by the
level surfaces of a time function is diffeomorphic to a product manifold RxΣ where Σ is a 3-dimensional manifold. Indeed,
the space-time can be parametrized by points on the slice t = 0 by following the integral curves of Dt. Moreover, relative
to this parametrization, the space-time metric takes the form
ds2 = -φ2(t, x)dt2 + gij(t, x) dxidxj , (8)
where x = (x1, x2, x3) are arbitrary coordinates on the slice t = 0. The function φ(t, x) is called the lapse function of
foliation; g is its first fundamental form. The three gauge conditions are gxt = gyt = gzt = 0.
The foliation is said to be normalized at infinity if Normal Foliation Condition
φ → 1 as x → ∞ on each leaf ∑t, (9)
The second fundamental form of the foliation, the extrinsic curvature of the leaves ∑t, is given by
kij = -(2φ)-1 ∂tgij . (10a)
and
tr k = 0, (10b)
is another gauge condition. Obviously, such a gauge choice is mathematically simple, but a physical explanation is not
provided. They define an initial data set (∑, g, k) with the property that the complement of a finite set in ä is
diffeomorphic to the complement of a ball in R3 (i.e., ∑ is diffeomorphic to R2 at infinity), and the notion of energy, and
of linear and angular momentum would be well defined and finite.
Then, they define that an initial data set (∑, g, k) satisfies the strongly asymptotically flat (S.A.F.) condition if g and k
are sufficiently smooth and there exists a coordinate system (x1,x2,x3) defined in a neighborhood of infinity such that, as r
= [
i
=
∑
1
3
(xi)2 ]1/2 → ∞. A S.A.F. Initial Data Set satisfies
gij = (1 + 2M/r)δij + o4(1/
r
3 2/
) (11a)
and
kij = o3(1/
r
2
) (11b)
The leading term (1 + 2M/r)δij will be called the Schwarzschild part of the metric g. Eq. (11) allows the static case kij =
0, but the interesting cases are, however, when kij ≠ 0.

Obviously, conditions (8) to (11) are satisfied by known static solutions. Thus, the S.A.F. initial data set is superficially
well-defined. However, the main problem is to show that S.A.F. initial data sets include dynamic situations. It is
physically meaningless to have such sets which include only static situations. Moreover, since form (8) together with
condition (10b) is a gauge choice, it is necessary to show such a choice is compatible with the equivalence principle, in
addition to condition (11). Then, to demonstrate that (11) is valid for dynamic situations, they should have provided a
general proof or, at least, a dynamic example.
A dynamic solution is a physical problem, and it is necessary to examine whether other physical requirements are
satisfied. In particular, it is crucial to show how a physical source is related.
The principle of causality requires that a radiation must have sources as the cause. Thus, to claim a solution as
dynamic, one must relate such a solution to a dynamic source. But, without addressing the physical source, Christodoulou
and Klainerman show only a possible existence of asymptotically flat maximal initial data sets. They introduce many
special mathematical constructions, involving long calculations, without giving a clear idea of how these building-blocks
will go together to eventually prove the main theorem.
Their argument is essentially based on mathematics that the equations
tr

g

k
= 0, and

∇

k
ij = 0 (12a)
are invariant with respect to the conformal transformation
g ij → Φ4gij , and k ij → Φ-2k ij . (12b)
where

∆
Φ -
1
8

R
Φ +

k


g
2
Φ-7 = 0, such that

R
=

k


g
2
. (12c)
But, the reviewer pointed out that the establishment of inequality (1.0.15) (see Appendix B) from eq. (12) is not clear. It
seems, this inequality should be related to the physical meaning of their construction. Also, the perceived existence of a
dynamic solution should have been established with an inequality.
Thus, the claimed dynamic nature has not been established. Since a distinct time-dependent initial condition has not
been shown, one cannot rule out the possibility that the class of S.A.F. initial data sets consists of only static solutions.
Being not an idealization of a realistic situation, a 梐ime-dependent” S. A. F. initial data set, if established, may not be
valid in physics unless such a solution is related to a dynamic source. In short, their claim of the existence of dynamic
solutions is invalid because simply it has not been proven.
Moreover, from the viewpoint of weak gravity [17,18], one would expect a S. A. F. initial data set to be compatible
with the field equation (3). The static isotopic solution is a S.A.F. initial data set, and its first order satisfies the static
linear equation,
1
2
∆
γ
ab = KT(m)ab . (13a)
Then, if a S.A.F. initial data set belongs to a physical solution, special relativity requires that the first order of a S.A.F.
initial data set satisfies the linear field eq. (3a) (see 3). But, solution (3b) suggests
kij = O(r-1), (13b)

which is incompatible with condition (11). Thus, a non-static S.A.F. initial data set cannot belong to a physical solution.
This means that the only physical S.A.F. initial data sets are the static solutions.
In general relativity an important question is: what does the gravitational field of a radiating asymptotically
Minkowskian system look like? The answer of this question, for the massive source, has already been given by eq. (3a).
On the other hand, Chrusciel, MacCallum, and Singleton [35] consider that the analysis of Christodoulou & Klainerman
suggested “that such systems generically do not satisfy the Bondi [32] -Penrose [36] -Sachs [37] asymptotic conditions”,
although Christodoulou and Klainerman made the claim of giving “rigorous derivation of the laws of gravitational
radiation proposed by Bondi [34]”.
5. THE INCOMPATIBILITY WITH EINSTEIN’S RADIATION FORMULA.
Since a dynamic situation necessarily leads to radiation [38], a crucial question is whether condition (11) is
compatible with Einstein’s radiation formula (in the Newtonian coupling constant κ) [18,30,33],
− = = =
∫









∑
dE
dt K Gta dSa
c
d Qij
dt ret
ij
12
55
3
3
1
3
2
( )
κ
> 0, (14a)
where
Qij qij ij q
= −
1
3
δ
, and
qij T xixjd x
=
∫
00 3
, (14b)
and δij = 1 or 0 if i ≠ j. Eq. (14) implies that a time average of G(2)tr (where r is the radial direction) is non-zero. Then,
to show the incompatibility, it is sufficient to show that a certain time average of G(1)ta , which does not involve the
second order derivative with respect to time, is zero in vacuum. It would be so if the metric g ab is periodic or almost
periodic. This is certainly the case for an isolated finite system.
For an isolated system consisting of particles with typical mass M, typical separation r, and typical velocities v,
Weinberg [18] estimated, the power radiated at a frequency of order v/ r will be of order P ≈ κ(v/r)6M2r4 or P ≈ Mv8/r
since κM/r is of order v2. The typical deacceleration arad of particles in the system owing to this energy loss is given by
the power P divided by the momentum Mv, or a rad ≈ v7/r. This may be compared with the accelerations computed in
Newtonian mechanics, which are of order v2/r, and with the post-Newtonian correction are of v4/r. Since radiation
reaction is smaller than the post-Newtonian effects by a factor v3, if v << c the velocity of light, the neglect of radiation
reaction is perfectly justified.
Consider, for instance, two particles of equal mass m with a circular orbit in the x-y plane whose origin is the center of
the circle. Thus, the metric gab(x, y, z, t’) is an almost periodic function of t’ (= t - r/c). Then, according to (9) and (11),
the time average of G(1)ta can be zero, and for a large distance r,
G(1)ta = - [∂a∂t(gkk) + ∂k∂tgak] ≈ o2(1/r7/2) + o3(1/r9/2) (a = x, y, or z). (15)

On the other hand, according to eq. (14), the existence of radiation implies that G(2)ta of O(1/r2) is non-zero. Thus,
according to eq. (2a), consistency with Gta = 0 can be achieved only if there is no radiation. This reaffirms the earlier
conclusion that the only valid S.A.F. initial data sets are the static solutions.
Apparently, their S.A.F. initial data sets cannot be extended to include (3b); otherwise this could have been done
already. Thus, in a way, their work further confirms the incompatibility between eq. (1a) and eq. (3a). Mathematically, the
existence of a dynamic S.A.F. initial data set for equation (1a) is also not possible because its incompatibility with eq. (3a)
(see Sections 3 & 4). Moreover, even if the requirement of relating to a dynamic source is ignored, the validity of their
conjecture that a “time-dependent” S.A.F. initial data set would exist mathematically, is unlikely since the plane-wave
solutions for eq. (1a) do not exist [8].
6. CONCLUSION AND DISCUSSION
Since Hulse-Taylor binary pulsar experiments support Einstein’s radiation formula [15], and thus support the
impossibility of having a dynamic solution for the Einstein equation (1a). Moreover, it has been shown that eq. (3a), as
the first order approximation, can be derived from the equivalence principle and related physical considerations which
lead to general relativity [27]. Concurrently, eq. (1a) has been proven to be an over simplification. Therefore, the need of
rectifying eq. (1a) is beyond doubt. Thus, from the viewpoint of physics, the static solutions are the only physical S.A.F.
initial data sets defined by Christodoulou & Klainerman.
Based on presumed time-dependency alone, they claimed that their solutions are dynamic developments of their initial
data sets. They also claimed of giving “a rigorous derivation of the laws of gravitational radiation proposed by Bondi
[34]”. But other theorists [35] consider that the analysis of Christodoulou & Klainerman suggested that the Bondi [34]
asymptotic conditions are not generically satisfied.
In pure mathematics, a given definition must be proven to be well-defined. The purpose is to ensure that this definition
is meaningful for the related theorems. But, Christodoulou and Klainerman have not shown any dynamic sources related
to their initial data sets, or provided a time-dependent example to illustrate the dynamic nature. Apparently, like many
other theorists who do not understand Einstein’s equivalence principle adequately [27] they believe that a time-dependent
solution must exist such that it is automatically a valid dynamic solution in physics. Such a crucial mistake is, however,
not an isolated problem in general relativity. For example, it was incorrectly believed that any form of energy were
equivalent to mass [39], that a Lorentz manifold were always physically valid, and that Fock’s [14] divergence were due
to his method because unbounded solutions were mistaken as bounded [27].
One might argue that they ignore the need of proving the dynamic nature of the strong asymptotically flat initial data
sets because they believe condition (12) to be valid in physics. However, since we are considering solutions of the Einstein
equation, we still have to show that condition (12) would indeed be an initial condition for a dynamic problem. Moreover,
a mathematical solution of a field equation may not necessarily be valid in physics. Thus, the claimed dynamic nature of
their solutions must be proved.
Besides, it is known that, for dynamic problems such as a simple two particle system, the Einstein equation has been
established as invalid [1,3,8,15,27,29]. Thus, even if their proof were valid for some situations, the 1915 field equation of
Einstein still have failed for dynamic problems. In other words, their of providing “a constructive proof of global, smooth,
nontrivial solutions to the Einstein-Vacuum equations”, cannot be valid in physics.

In general relativity, a weak gravitational radiation from an isolated source, is related to γab ≈ Aab(t-r)/r. (The non-
existence of bounded plane-waves which should be related to Aab(t-r), manifests there is no dynamic solution for the 1915
equation.) Although one could assume the possibility of having a mathematical solution in vacuum which does not consist
of a function Aab(t-r)/r, such a solution would not be possible to be generated by an isolated source since some first order
terms have been excluded. Thus, to show that a time-dependent solution, which is generated by mathematical
manipulations unrelated to physics, is a dynamic solution, it is necessary (as in the case of electrodynamics) to relate such
a solution with a dynamic source.
Thus, the proof of Christodoulou and Klainerman is, at least, incomplete; and is invalid because their assumption is
incompatible with the linear equation for weak gravity and Einstein’s radiation formula. On the other hand, if their work
means the impossibility of including the linear solutions to their initial sets, their work would be an additional
confirmation of the incompatibility between eqs. (1a) and (3a).
It is unfortunately that many well-known theorists were unable to distinguish the physical among mathematical
solutions. In particular, due to inadequate understanding of physics, Synge [40] earlier and Friedman [41] currently
advocated that “the existence of local Minkowski space has replaced the equivalence principle that initially motivated
it”1). This is incorrect since a Lorentz manifold may not even be diffeomorphic to a physical space [22]. Experimentally,
there are many evidences that a valid space-time coordinate system must be restricted. In particular, the observed
gravitational red shifts unequivocally imply that the existence of local Minkowski space alone is inadequate for a
satisfaction of the equivalence principle. Due to Einstein’s principle has not been adequately understood [24,27], physical
requirements are often ignored [1,8,9,22,27]. It seems, Christodoulou and Klainerman are not the exceptions.
Physics includes and depends on the usage of mathematics. But, satisfying a field equation is only one physical
requirement among many [27]. As Einstein [25,32] demonstrated, understanding the physical principles and special
relativity in particular, is crucial in general relativity [7-9,15,27,39]. Perhaps, it would be beneficial for those authors who
utilize the work of Christodoulou and Klainerman, to review their book in connection with the author’s own papers in
details. It is important to understand general relativity from the viewpoint of physics as Weinberg [18] advocates.
Acknowledgments
This paper is dedicated to late Professor H. W. Ellis and Professor R. R. D. Kemp, Department of Mathematics,
Queen’s University, Ontario, Canada. Stimulating discussions with Professor John L. Friedman, Professor J. E. Hogarth,
Professor P. Morrison are gratefully acknowledged. Special thanks are to the referees for valuable comments and useful
suggestions, and Ms. Sunny. Auyang for help in finding useful references, Ms. P. Ma for the French translation, and Mr.
Ming Wong for reading through the manuscript.
This work is supported in part by Innotec Design, Inc.
APPENDIX A: A BOOK REVIEW ON “THE GLOBAL NONLINEAR STABILITY OF THE MINKOWSKI
SPACE”
This book review by Volker Perlick originally appeared in ZfM [42] in 1996; and, with the kind permission of its
Editor, B. Uegner, will be republished in the journal, GRG [43] again with the editorial note, “one may extract two
messages: on the one hand, (by seeing e.g. how often this book has been cited), the result is in fact interesting even today,
and on the other hand: There exists, up to now no generally understandable proof of it.” However, for the Editorial Board
of Classical & Quantum Gravity, their strong faith on this book, in particular their claimed proof on the existence of

radiative bounded solutions, remains unchanged [44]. For the convenience of the reader, this review is provided as an
appendix. The review is as follows:
“For Einstein’s vacuum field equation, it is a difficult task to investigate the existence of solutions with prescribed
global properties. A very interesting result on that score is the topic of the book under review. The authors prove the
existence of globally hyperbolic, geodesically complete, and asymptotically flat solutions that are close to (but different
from) Minkowski space. These solutions are constructed by solving the initial value problem associated with Einstein’s
vacuum field equation. More precisely, the main theorem of the book says that any initial data, given on R3, that are
asymptotically flat and sufficiently close to the data for Minkowski space give rise to a solution with the desired
properties. In physical terms, these solutions can be interpreted as spacetimes filled with source-free gravitational
radiation. Geodesic completeness means that there are no singularities. At first sight, this theorem might appear
intuitively obvious and the enormous amount of work necessary for the proof might come as a surprise. The following two
facts, however, should caution everyone against such an attitude. First, it is known that there are nonlinear hyperbolic
partial differential equations (e.g., the equation of motion for waves in non-linear elastic media) for which even arbitrarily
small localized initial data lead to singularities. Second, all earlier attempts to find geodesically complete and
asymptotically flat solutions of Einstein’s vacuum equation other than Minkowski space had failed. In the class of
spherically symmetric spacetime and in the class of static spacetimes the existence of such solutions is even excluded by
classical theorems. These facts indicate that the theorem is, indeed, highly non-trivial. Yet even in the light of these facts
it is still amazing that the proof of the theorem fills a book of about 500 pages. To a large part, the methods needed for the
proof are rather elementary; abstract methods from functional analysis are used only in so far as a lot of L2 norms have to
be estimated. What makes the proof involved and difficult to follow is that the authors introduce many special
mathematical constructions, involving long calculations, without giving a clear idea of how these building-blocks will go
together to eventually prove the theorem. The introduction, almost 30 pages long, is of little help in this respect. Whereas
giving a good idea of the problems to be faced and of the basic tools necessary to overcome each problem, the introduction
sheds no light on the line of thought along which the proof will proceed. For mathematical details without seeing the
thread of the story. This is exactly what happened to the reviewer.”
“To give at least a vague idea of how the desired solutions of Einstein’s vacuum equation are constructed, let us
mention that each solution comes with the following: (a) a maximal spacelike foliation generalizing the standard foliation
into surfaces t = const. in Minkowski space; (b) a so-called optical function u, i.e. a solution u of the eikonal equation that
generalizes the outgoing null function u = r - t on Minkowski space; (c) a family of “almost conformal killing vector fields
on Minkowski space. The construction of these objects and the study of their properties requires a lot of technicalities.
Another important tool is the study of “Bianchi equations” for “Weyl tensor fields”. By definition, a Weyl tensor field is a
fourth rank tensor field that satisfies the algebraic identities of the conformal curvature tensor, and Bianchi equations are
generalizations of the differential Bianchi identities.”
“In addition to the difficulties that are in the nature of the matter the reader has to struggle with a lot of unnecessary
problems caused by inaccurate formulations and misprints. E.g., “Theorem 1.0.2” is not a theorem but rather an
inaccurately phrased definition. The principle of conservation of signature” presented on p. 148 looks like a mathematical
theorem that should be proved; instead, it is advertised as an “heuristic principle which is essentially self-evident.” For all
these reasons, reading this book is not exactly great fun. Probably only very few readers are willing to struggle through
these 500 pages to verify the proof of just one single theorem, however interesting.”

“Before this book appeared in 1993 its content was already circulating in the relativity community in form of a preprint
that gained some notoriety for being extremely voluminous and extremely hard to read. Unfortunately, any hope that final
version would be easier to digest is now disappointed. Nonetheless, it is to be emphasized that the result presented in this
book is very important. Therefore, any one interested in relativity and/or in nonlinear partial differential equations is
recommended to read at least the introduction.
Volker Perlick
Institut f. Theor. Physik, TU Berlin
10623 Berlin, Germany
Author’s note: This review actually suggests that problems would be adequately identified in the introduction. As shown
in the present paper, the possible nonexistence of their dynamic solutions and its incompatibility with Einstein’s radiation
formula can be discovered in their introduction. Their book has often been cited [45-57], in spite of the invalid “proof”.
Note, however, such citations in some journals have stopped since 1996.
APPENDIX B: THE SMALLNESS ASSUMPTION AND THE S.A.F INITIAL DATA CONDITION
In this Appendix B, it is pointed out that a dynamic S.A.F. condition need not necessarily exist. Also, it is strange that
the“physical” solutions are constructed with only mathematical considerations but without appropriate physical
considerations. In their book, Christodoulou and Klainerman wrote:
“Our construction requires initial data sets that satisfy, in addition to the constraint equations, the maximal condition
tr k = 0. We will refer to them as maximal in what follows:”
“To make the statement of our main theorem precise, we need also to define what we mean by the global smallness
assumption. Before stating this condition, we assume the metric g to be complete and we introduce the following quantity:
Q(x(0), b) =
Sup
Σ
{b-2(d02 + b2)3Ric2}
+ b-3{
(
l
=
∑
∫
0
3
Σ
d02 + b2)l+1∇lk2 +
(
l
=
∑
∫
0
3
Σ
d02 + b2)l+3∇lB2}
where d0(x) = d(x(0),x) is the Riemannian geodesic distance between the point x and a given point x(0) on Σ, b is a positive
constant, Ric2 = RijRij, ∇l denotes the 1-covariant derivatives, and B is the symmetric, traceless 2-tensor tensor
Bij = ∈jab∇a(Rib - ¼ gibR).
The symmetry and traceless of B follow immediately from the twice-contracted Bianchi identities ∇jRij - ½ ∇i R = 0. In
the fact we can write
Bij = (1/2)(∈iab∇a

R
jb + ∈jab∇a

R
ib)
where

R
ij is the traceless part of Rij, Rij =

R
ij + 1/3 R gij.
Theorem 1.0.2 (The Global Smallness Assumption) We say that an S.A.F initial data set, ( Σ, g, k), satisfies the global
smallness assumption if the metric g is complete and there exists a sufficiently small positive ∈ such that

Inf
x b
( )
,
0
0
∈ ≥Σ
Q(x(0), b) < ∈ (1.0.15)
Theorem 1.0.3 (Second Version of the Main Theorem) Any strongly asymptotically flat , Maximal, initial data set that
satisfies the global smallness assumption 1.0.15 leads to a unique, globally hyperbolic, smooth, and geodesically complete
solution of the Einstein-Vacuum equation foliated by a normal maximal time foliation. Moreover, this development is
globally asymptotically flat.
Remark 1. In view of the scale invariance property of the Einstein-Vacuum equations, any initial data set Σ, g, k for
which Q(x0, b) < ∈ can be rescaled to the new initial data set ä, g’, k’ with g’ = b-2g, k’ = b-1k for which Q(x0, 1) < ∈.
The global existence for the new set is equivalent to the global existence for the original set. This is due to the fact that the
developments g, g’ of the two sets are related by g’ = b-2g. It thus suffices to prove the theorem under the global smallness
assumption
Inf
x
( )0
∈Σ
Q(x(0), b) < ∈.”
Then, they prove that for given arbitrary solutions
~
g
,
~
k
to the equations
tr
g
~
~
k
= 0, (1.0.16a)
~~
∇
j
ji
k
= 0 (1.0.16b)
which are invariant with respect to the conformal transformation, this suffices to insure an initial data set (Σ, g, k)
satisfying the S.A.F. condition if
~
g
ij = δij +
o r
4
3 2
( )
/
−
, and
~
k
ij =
o r
3
5/2
( )
−
.
and the negative part of

R
satisfies the smallness condition. Moreover, g and k satisfy the global smallness assumption
of the theorem provided that the metric
~
g
is complete and that there exists a mall positive ∈ such that
Inf
x a( ) ,0 0
∈ ≥Σ
{
Sup
Σ
(
d
02 + a2)3

Ric
~

2}
+
(
l
=
∑
∫
0
3
Σ
d
02 + a2)l+2
~~
∇

Ric
2 +
(
l
=
∑
∫
0
3
Σ
d
02 + a2)l+1
~~
∇

k
2 < ∈
where
d
0(x) (=
d
0) denotes the Riemannian geodesic distance relative to
~
g
between the point x and a given point x(0)
on Σ. Thus, it remains to discuss whether the equation 1.0.16a and 1.0.16b have solutions.
Author’s note: However, because condition (1.0.15), (11) and equation (1.0.16) have no dynamic requirements, in
their proofs, there is no assurance for the existence of a dynamic S.A.F. initial data set.
Endnote
1) The implicit reduction of Einstein’s equivalence principle to be just the signature of Lorentz metric is clarified by
using the word “replaced” in Friedman’s [41] remark. Thus, the theory of the Wheeler-Hawking [17,58] school is not

really general relativity. In usual practice, when an assumption in physics is modified, the modification is explicitly
clarified with adequate reasoning. An explanation of this unusual practice of the Wheeler-Hawking school would be
that they did not understand Einstein’s equivalence principle adequately and therefore were not aware of that this
reinterpretation is actually a reduction until recently. This would explain also why Einstein’s equivalence principle
was not explained adequately by this school [17,26,30,58-60]. Recently, it has been proven that their reduction is
inconsistent with Einstein’s own interpretation and physical principles [23,27] as well as in disagreement with
experiments including the Michelson-Morley experiment [61]. Nevertheless, the advocates disregard all these
inconsistencies by claiming that a coordinate system (including its metric) has no physical meaning. Apparently, they
did not see that the observed gravitational red shifts unequivocally imply that their reduction (or reinterpretation) is
invalid in physics.
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