Note that the Einstein radiation formula depends on (3) as a first-order approximation. Thus, metric g(( must be bounded. Otherwise G(( = 0 can be satisfied. For example, the metric of Bondi et al. [15] is
ds2 = exp(2()(d( 2 - d(2) - u2(ch2( (d(2 + d(2) + sh2( cos2( (d(2 - d(2) - 2sh2( sin2( d(d((, (8)
where (, (, ( are functions of u (= ( - ( ). It satisfies the differential equation (i.e., their eq. (2.8(),
2(' = u(('2 + ('2 sh2(2). (9)
However, metric (8) is not bounded, because this would require the impossibility of u2 < constant. Note that an unbounded function of u, f(u) grows anomaly large as time ( goes by.
It should be noted also that metric (8) is only a plane, but not a periodic function because a smooth periodic function must be bounded. This unboundedness is a symptom of unphysical solutions because they cannot be related to a dynamic source (see also [9,11]). Note that solution (8) can be used to construct a smooth one-parameter family of solutions [11] although solution (8) is incompatible with Einstein's notion of weak gravity [2].
In 1953, questions were raised by Schiedigger [30] as to whether gravitational radiation has any well-defined existence. The failure of recognizing G(( = 0 as invalid for gravitational waves is due to mistaking (3) as a first-order approximation of (1). Thus, in spite of Einstein's discovery [3] and Hogarth's conjecture6) [31] on the need of modification, the incompatibility between (1) and (3) was not proven until 1993 [13] after the non-existence of the plane-waves for G(( = 0, has been proven [9,18].
4. Gravitational Radiation and the 1995 update of the Einstein Equation
In general, (3) is actually an approximation of the 1995 update of the Einstein equation [13],
G(( ( R(( - g((R = - K (T(m)(( - t(g)(((, (10)
where t(g)(( is the energy-stress tensors for gravity. Then,
((T(m)(( = 0, and ((t(g)(( = 0. (11)
Equation (11) implies that the equivalence principle would be satisfied. From (10), the equation in vacuum is
G(( ( R(( - g((R = K t(g)(( . (10')
Note that t(g)(( is equivalent to G(2)(( (and Einstein's gravitational pseudotensor) in terms of his radiation formula. The fact that t(g)(( and G(2)(( are related under some circumstances does not cause G(2)(( to be an energy-stress nor t(g)(( a geometric part, just as G(( and T(( must be considered as distinct in (1).
When gravitational wave is present, the gravitational energy-stress tensor t(g)(( is non-zero. Thus, a gravitational radiation does carry energy-momentum as physics requires. This explains also that the absence of an anti-gravity coupling which is determined by Einstein's radiation formula, is the physical reason that the 1915 Einstein equation (1) is incompatible with radiation.
Note that the radiation of the binary pulsar can be calculated without detailed knowledge of t(g)((. From (10'), the approximate value of t(g)(( at vacuum can be calculated through G((/K as before since the first-order approximation of g(( can be calculated through (3). In view of the facts that Kt(g) (( is of the fifth order in a post-Newtonian approximation, that the deceleration due to radiation is of the three and a half order in a post-Newtonian approximation [8] and that the perihelion of Mercury was successfully calculated with the second-order approximation from (1), the orbits of the binary pulsar can be calculated with the second-order post-Newtonian approximation of (10) by using (1) (see also Section 6). Thus, the calculation approaches of Damour and Taylor [25,26] would be essentially valid except that they did not realize the crucial fact that (3) is actually an approximation of the update equation (10) [13].
In light of the above, the Hulse-Taylor experiments support the anti-gravity coupling being crucial to the existence of the gravitational wave [10,13], and (3) being an approximation of weak waves generated by massive matter. Thus, it has been experimentally verified that (1) is incompatible with radiation.
It should be noted also that the existence of an anti-gravity coupling7) means the energy conditions in the singularity theorems [6,17] are not valid at least for a dynamic situation. Thus, the existence of singularity is not certain, and the claim of inevitably breaking of general relativity is actually baseless since these singularity theorems have been proven to be unrealistic in physics. As pointed out by Einstein [2], his equation may not be valid for very high density of field and matter. In short, the singularity theorems show only the breaking down of theories of the Wheeler-Hawking school, which are actually different3) from general relativity.
The theories of this school, in addition to making crucial mistakes in mathematics as shown in this paper (see also [11,28]), differ from general relativity in at least the following important aspects:
1) They reject an anti-gravity coupling7), which is considered as highly probable by Einstein himself.
2) They implicitly replaced Einstein's equivalence principle in physics3) with merely the mathematical requirement of the existence of local Minkowski spaces [5,6].
3) They, do not consider physical principles [9-11,28] (see also Section 5), such as the principle of causality, the coordinate relativistic causality, the correspondence principle and etc. of which the satisfaction is vital for a physical space, which models reality, such that Einstein's equivalence principle can be applicable.
Thus, in spite of currently declaring their theories as the development of general relativity, these theories actually contradict crucial features that are indispensable in Einstein's theory of general relativity. More importantly, in the development of their so-called "orthodox theory," they implicitly violate physical principles that took generations to establish. As a result, Einstein's theory has been unfairly considered as irrelevant in the eyes of many physicists.
Of course, the exact form of t(g)(( is important for the investigation of high density of field. However, it seems, the physics of very high density of field and matter is not yet mature enough at present to allow a definitive conclusion. For instance, it is unclear what influence the discovery of quarks and gluons in particle physics would have on the evolution of stars. It is known that atomic physics supports the notion of white-dwarf stars, and that nuclear physics leads to the notion of neutron stars.
5. Physically Invalid Unbounded "Gravitational Waves" and the Principle of Causality
"To my mind there must be at the bottom of it all, not an equation, but an utterly simple idea. And to me that idea, when we finally discover it, will be so compelling, so inevitable, that we will say to one another, 'Oh, how beautiful. How could it have been otherwise?' " -- J. A. Wheeler [32].
It seems, the principle of causality2) (i.e., phenomena can be explained in terms of identifiable causes) [9,10] would be qualified as Wheeler's utterly simple idea. Being a physicist, his notion of beauty should be based on compelling and inevitability, but would not be based on some perceived mathematical ideas. It will be shown that the principle of causality is useful in examining validity of accepted "wave" solutions.
According to the principle of causality, a wave solution must be related to a dynamic source, and therefore is not just a time-dependent metric. A time-dependent solution, which can be obtained simply by a coordinate transformation, may not be related to a dynamic source8) [33]. Even in electrodynamics, satisfying the vacuum equation can be insufficient. For instance, the electromagnetic potential solution A0[exp(t - z)2] (A0 is a constant), is not valid in physics because one cannot relate such a solution to a dynamic source. Thus, as shown in Section 4, a solution free of singularities may not be physically valid.
A major problem in general relativity is that the equivalence principle has not been understood adequately [11,34]. Since a Lorentz manifold was mistaken as always valid, physical principles were often not considered. For instance, the principle of causality was neglected such that a gravitational wave was not considered as related to a dynamic source, which may not be just the source term in the field equation [8,35].
Since the principle of causality was not understood adequately, solutions with arbitrary nonphysical parameters were accepted as valid [34]. Similarly, Misner, Thorne & Wheeler [5], assumed that the metric due to an electromagnetic plane-wave is invariant with respect to a rotation whose axis is in the direction of propagation. Consequently, in addition to the fact that the polarization is incorrect, Misner et al. were not aware of that, in disagreement with what they stated, such a metric cannot be bounded. Such unbounded solutions disagree with experiments [10,11].
Among the existing so-called wave solutions, not only Einstein's equivalence principle but the principle of causality is not satisfied because they cannot be related to a dynamic source. (However, a source term in an equation, though related to, may not necessarily represent the physical cause [9,34].) Here, examples of accepted "gravitational waves" are shown as actually invalid in physics.
1. Let us examine the cylindrical waves of Einstein & Rosen [29]. In cylindrical coordinates, (, (, and z,
ds2 = exp(2( - 2()(dT2 - d(2) - (2exp(-2()d(2 - exp(2()dz2 (12)
where T is the time coordinate, and ( and ( are functions of ( and T. They satisfy the equations
((( + (1/()(( - (TT = 0, (( = ([((2 + (T2], and (T = 2((((T. (13)
Rosen [36] consider the energy-stress tensor T(( that has cylindrical symmetry. He found that
T44 + t44 = 0, and T4l + t4l = 0 (14)
where t(( is Einstein's gravitational pseudotensor, t4l is momentum in the radial direction.
However, Weber & Wheeler [37] argued that these results are meaningless since t(( is not a tensor. They further pointed out that the wave is unbounded and therefore the energy is undefined. Moreover, they claimed metric (12) satisfying the equivalence principle and speculated that the energy flux is non-zero.
Their claim shows an inadequate understanding of the equivalence principle. To satisfy this principle requires that a time-like geodesic must represent a physical free fall. This means that all (not just some) physical requirements are necessarily satisfied. Thus, the equivalence principle may not be satisfied in a Lorentz Manifold [11,35], which implies only the necessary condition of the mathematical existence of a co-moving local Minkowski space along a time-like geodesic. It will be shown that manifold (12) cannot satisfy coordinate relativistic causality. Moreover, as pointed out earlier, an unbounded wave is unphysical.
Weber and Wheeler's arguments for unboundedness are complicated, and they agreed with Fierz's analysis, based on (13), that ( is a strictly positive where ( = 0 [37]. However, it is possible to see that there is no physical wave solution in a simpler way. Gravitational red shifts imply that gtt ( 1 [2]; and
-g(( ( gtt , -g((/(2 ( gtt , and -gzz ( gtt , (15a)
are implies by coordinate relativistic causality. Thus, according to these constraints, from metric (12) one has
exp(2() ( 1 and exp (2() ( exp(4(). (15b)
Equation (15) implies that gtt ( 1 and that ( ( 0. However, this also means that the condition ( > 0 cannot be met. Thus, this shows again that there is no physical wave solution for G(( = 0.
Weber and Wheeler are probably the earliest to show the unboundedness of a wave solution for G(( = 0. Nevertheless, due to their inadequate understanding of the equivalence principle, they did not reach a valid conclusion. It is ironic that they therefore criticized Rosen who come to a valid conclusion, though with dubious reasoning.
2. Robinson and Trautman [38] dealt with a metric of spherical "gravitational waves" for G(( = 0. However, their metric has the same problem of unboundedness and having no dynamic source connection. This confirms further that the cause of this problem is intrinsically physical in nature. Their metric has the following form:
ds2 = 2d(d( + (K - 2H( - 2m/()d(2 - (2p-2{[d( + ((q/(()d(]2 + [d( +((q/(()d(]2}, (16a)
where m is a function of ( only, p and q are functions of (, (, and (,
H = p-1(p/(( + p(2p-1q/(((( - pq (2p-1/(((( , (16b)
and K is the Gaussian curvature of the surface ( = 1, ( = constant,
K = p2((2/((2 + (2/((2)ln p. (16c)
For this metric, the empty-space condition G(( = 0 reduces to
(2q/((2 + (2q/((2 = 0, and (2K/((2 + (2K/((2 = 4p-2((/(( - 3H)m. (17)
To see this metric has no dynamic connection, let us examine their special case as follows:
ds2 = 2d(d( - 2Hd(2 - d(2 - d(2, and (H/(( = (2H/((2 + (2H/((2 = 0. (18)
This is a plane-fronted "wave" [39] derived from metric (16) by specializing
p = 1 + ((2 + (2)K(()/4. (19a)
substituting
( = (-2 + (-1, ( = (, ( = (2, ( = (2, q = (4, (19b)
where ( is constant, and taking the limit as ( tends to zero [38]. Although (18) is a Lorentz metric, there is a singularity on every wave front where the homogeneity conditions
(3H/((3 = (3H/((3 = 0. (20)
are violated [38]. Obviously, this is also incompatible with Einstein's notion of weak gravity [2]. A problem in current theory is its rather insensitivity toward theoretical self-consistency [9,13,35,40-42].
3. To illustrate the non-existence of a bounded radiating physical solution further, let us examine a recent solution of R(( = 0, the cylindrical symmetry solution of Au, Fang & To [43]. Their metric is
ds2 = N2(c2dt2 - dz2) - L2d(2 - M2(2d(2 (21)
where
N2 = (-4exp(-4((d() exp(2n1), L2 = (-8(1 + (()2exp(-6((d(),
and
M2 = exp(2((d() where n1= n1(ct - z), and ( = ((()
are respectively arbitrary functions of (ct - z) and of (. The function n1(ct - z) makes N2 a propagating wave. If solution (21) were a physical solution, M should be a bounded function of (, i.e.,
exp(2((d() < C12 (22)
for some constant C1. However, this also means that N is not bounded for small (. Moreover, if light velocity is not larger than its vacuum velocity c, one should have N2/L2 and N2/M2 ( 1. It thus follows that
(1 + (()2 ( (4 exp(2((d()exp(2n1), and exp(6((d() ( exp(2n1) (-4. (23)
Hence,
(1/( + ()2 ( (2/3 exp (8n1/3) and (2 > ( O((2/3). (24)
Thus, condition (24) is also inconsistent with condition (22). In summary, solution (21) is also not a physical solution and is unbounded in contrast to as required by the principle of causality.
4. To illustrate an invalid source and an intrinsic non-physical space, consider the following metric ,
ds2 = du dv + Hdu2 - dxi dxi, where H = hij(u)xi xj (25)
where u = ct - z, v = ct + z, x = x1 and y = x2, hii(u) ( 0, and hij = hji [44]. This metric satisfies the harmonic gauge. The cause of metric (25) can be an electromagnetic plane wave. Metric (25) satisfies
((( (((( (tt = -2{hxx(u) + hyy(u)} where ((( = g(( - (((. (26)
However, this does not mean that causality is satisfied although metric (25) is related to a dynamic source. It will be shown that (25) is not a physical solution because physical principles are violated.
A light trajectory satisfies ds2 = 0 [2]. For a light in the z-direction (i.e. dx = dy = 0), one obtains
dz/dt = c or -c (1 + H)/(1 - H); but H ( 0 (27)
would fail since hii(u) ( 0 ; and so coordinate relativistic causality would also fail. Thus, a formal satisfaction of the conservation law due to ((G(( ( 0, is inadequate to ensure the validity of (1).
Moreover, the gravitational force is related to (ztt = (1/2)(H/(t. There are arbitrary non-physical parameters (the choice of origin) that are unrelated to the cause (a plane wave). Apparently, believing that any Lorentz manifold is valid in physics, Penrose [44] over-looked the physical requirements, in particular the principle of causality. Experimentally, being unbounded, metric (25) is also incompatible with the calculation of light bending and classical electrodynamics.
These examples confirm that there is no bounded wave solution for (1) although a "time-dependent" solution need not be logarithmic divergent [14]. A fundamental reason for the boundedness of a dynamic solution for gravity, is the equivalence principle [11]. This would mean that the hyperboloid solution in Friedmann's theory might not be valid in general relativity (see Appendix).
6. Conclusions and Discussions
In general relativity, the existence of gravitational wave is a crucial test of the field equation. Thus, an important question is: what does the gravitational field of a radiating asymptotically Minkowskian system look like? Without experimental inputs, to answer this question would be very difficult.
Einstein [2] proposed the linearized theory for a weak radiating gravitational field. But, Bondi [24] commented, "it is never entirely clear whether solutions derived by the usual method of linear approximation necessarily correspond in every case to exact solutions, or whether there might be spurious linear solutions which are not in any sense approximations to exact ones." Thus, in calculating gravitational waves from the Einstein equation, problems are considered as due to the method rather than inherent in the equations.
Physically, it is natural to continue assuming Einstein's notion of weak gravity is valid. (Boundedness, though a physical requirement, may not be mathematically compatible to a nonlinear field equation. But, no one except perhaps Gullstrand [40,41], expected the nonexistence of dynamic solutions.) The complexity of the Einstein equation makes it very difficult to have a close form. Thus, it is necessary that a method of expansion should be used to examine the problem of weak gravity, if one expects such an expansion to be valid.
A factor which contributes to this faith is that ((G(( ( 0 implies ((T(m)(( = 0, the energy-momentum conservation law. However, this is only necessary but not sufficient for a dynamic solution. Although the 1915 equation gives an excellent description of planetary motion, including the advance of the perihelion of Mercury, this is essentially a test-particle theory, in which the reaction of the test particle is neglected. Thus, the so obtained solutions are not dynamic solutions. As pointed out by Gullstrand [41,45] such a solution may not be obtainable as a limit of a dynamic solution. Nevertheless, Einstein, Infeld, and Hoffmann [22] incorrectly assumed the existence of bounded dynamic solution and deduced the geodesic equation from the 1915 equation. Recently, Feymann [23] made the same incorrect assumption that a physical requirement would be unconditionally applicable to a mathematical equation.
The nonlinear nature of Einstein equation certainly gives surprises. In 1959, Fock [14] pointed out that, in harmonic coordinates, there are divergent logarithmic deviations from expected linearized behavior of the radiation. After the discovery that some vacuum solutions are not logarithmic divergent [15], the inadequacy of Einstein's equation was not recognized. Instead, the method of calculation was mistaken as the problem.
To avoid the appearance of logarithms, Bondi et al. [24] and Sachs [46] introduced a new approach to gravitational radiation theory. They used a special type of coordinate system, and instead of assuming an asymptotic expansion in the gravitational coupling constant (, they assume the existence of an asymptotic expansion in inverse power of the distance r (from the origin where the isolated source is located in r ( a, which is a positive constant). The approach of Bondi-Sachs was clarified by the geometrical 'conformal' reformulation of Penrose [47].
However, this approach is unsatisfactory, "because it rests on a set of assumptions that have not been shown to be satisfied by a sufficiently general solution of the inhomogeneous Einstein field equation [48]." In other words, this approach provides only a definition of a class of space-times that one would like to associate to radiative isolated systems, neither the global consistency nor the physical appropriateness of this definition has been proven. Moreover, perturbation calculations have given some hints of inconsistency between the Bondi-Sachs-Penrose definition and some approximate solution of the field equation. Not less important, it seems a priori difficult to relate to the source located within r ( a [48].
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