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Applied and Pure Research Institute
17 Newcastle Drive, Nashua, NH 03060
Physics Essays, vol. 13, no. 4, 2000
Abstract
It is shown that the 1915 Einstein equation is incompatible with the physical notion that a wave carries away energy-momentum. This proof is compatible with that Maxwell-Newton Approximation (the linear field equation for weak gravity), and is supported by the binary pulsar experiments. For dynamic problems, the linear field equation is independent of, and furthermore incompatible with the Einstein equation. The linear equation, as a first-order approximation, requires the existence of the weak gravitational wave such that it must be bounded in amplitude and be related to the dynamics of the source of radiation. Due to neglecting these crucial physical associations, in addition to inadequate understanding of the equivalence principle, unphysical solutions were mistaken as gravitational waves. It is concluded theoretically that, as Einstein and Rosen suggested, a physical gravitational wave solution for the 1915 equation does not exist. This conclusion is given further supports by analyzing the issue of plane-waves versus exact "wave" solutions. Moreover, the approaches of Damour and Taylor for the radiation of binary pulsars would be valid only if they are as an approximation of the equation of 1995 update. In addition, the update equation shows that the singularity theorems prove only the breaking down of Wheeler-Hawking theories, but not general relativity. It is pointed out that some Lorentz manifolds are among those that actually disagree with known experimental facts.
Key Words: compatibility, dynamic solution, gravitational radiation, principle of causality, plane-wave, Wheeler-Hawking theories
1. Introduction
In physics, the existence of a wave is due to the fact, as required by special relativity, that a physical cause must propagate with a finite speed [1]. This implies also that a wave carries energy-momentum. Thus, the field equation for gravity must be able to accommodate the gravitational wave, which carries away gravitational energy-momentum. In this paper, it will be shown that the Einstein equation of 1915 fails this.
In general relativity, the Einstein equation of 1915 [2] for gravity of space-time metric g(( is
G(( ( R(( - g((R = - KT (m)(( , (1)
where G(( is the Einstein tensor, R(( is the Ricci curvature tensor, T(m)(( is the energy-stress tensor for massive matter, and K (= 8((c-2, and ( is the Newtonian coupling constant) is the coupling constant1). Thus,
G(( ( R(( - g((R = 0, or R(( = 0, (1')
at vacuum. However, (1') also implies no gravitational wave to carry away energy-momentum.
An incompatibility with radiation was first discovered by Einstein & Rosen [3,4] in 1936. However, due to conceptual and mathematical errors then, their discovery was not accepted. These errors form the basis of the so-called geometric viewpoint of the Wheeler-Hawking school [5,6] (see also Section 4). An obvious problem of their viewpoint is that one cannot distinguish a physical solution among mathematical solutions [7].
Conceptually, one would argue incorrectly that (1') carries energy-momentum because
G(( ( G(1)(( + G(2)(( (2a)
where G(1)(( consists of the linear terms (of the deviation ((( = g(( - ((( from the flat metric ((() in G(( , and G(2)(( consists of the others. Since G(2)(( has been identified as equivalent to the gravitational energy-stress of Einstein's notion [8], it seemed obvious that G(2)(( carries the energy-momentum. However, unless (1) can accommodate a physical gravitational wave, such an argument has no meaning. Moreover, no wave solution has ever been obtained for equation (1). In fact, this is impossible (see Section 2).
There are so-called "wave solutions" for (1'), but they are actually invalid in physics (see §§ 3 & 5) since physical requirements (such as the principle of causality2), the equivalence principle, and so on) are not satisfied. In fact, some of them have been proven to be in disagreement with experiments [9,10]. Their invalid acceptance is due to the incorrect belief3) that the equivalence principle were satisfied by any Lorentz manifold [11].
Moreover, Einstein's notion cannot be exact, since it is not localizable [12]. In a field theory, a central problem is the exchange of energy between a particle and the field where the particle is located [13]. Therefore, the gravitational energy-stress must be a tensor (see also Section 4).
2. The Gravitational Wave and Nonexistence of Dynamic Solutions for Einstein's Equation
First, a major problem is a mathematical error on the relationship between (1) and its "linearization". It was incorrectly believed that the linear Maxwell-Newton Approximation [13]
( c(c(( = - K T(m) (( , where (( = ((( - (((((cd(cd) (3a)
and
(((xi, t) = - (T(((yi, (t - R)]d3y, where R2 =(xi - yi)2 . (3b)
always provides the first-order approximation for equation (1). This belief was verified for the static case only.
For a dynamic4) case, however, this is no longer valid. While the Cauchy data can be arbitrary for (3a), but not for (1). The Cauchy data of (1) must satisfy four constraint equations, G(t = -KT(m)(t (( = x, y, z, t) since G(t contains only first-order time derivatives [8]. This shows that (3a) would be dynamically incompatible5) with equation (1) [10]. Further analysis shows that, in terms of both theory [11] and experiments [13], this mathematical incompatibility is in favor of (3), instead of (1).
In 1957, Fock [14] pointed out that, in harmonic coordinates, there are divergent logarithmic deviations from expected linearized behavior of the radiation. This was interpreted to mean merely that the contribution of the complicated nonlinear terms in the Einstein equation cannot be dealt with satisfactorily following this method and that other approach is needed. Subsequently, vacuum solutions that do not involve logarithmic deviation, were founded by Bondi, Pirani & Robinson [15] in 1959. Thus, the incorrect interpretation appears to be justified and the faith on the dynamic solutions maintained. It was not recognized until 1995 [13] that such a symptom of divergence actually shows the absence of bounded physical dynamic solutions.
In physics, the amplitude of a wave is generally related to its energy density and its source. Equation (3) shows that a gravitational wave is bounded and is related to the dynamic of the source. These are useful to prove that (3), as the first-order approximation for a dynamic problem, is incompatible with equation (1). Its existing "wave" solutions are unbounded and therefore cannot be associated with a dynamic source [11]. In other words, there is no evidence for the existence of a physical dynamic solution.
With the Hulse-Taylor binary pulsar experiment [16], it became easier to identify that the problem is in (1). Subsequently, it has been shown that (3), as a first-order approximation, can be derived from physical requirements which lead to general relativity [11]. Thus, (3) is on solid theoretical ground and general relativity remains a viable theory. Note, however, that the proof of the nonexistence of bounded dynamic solutions for (1) is essentially independent of the experimental supports for (3).
To prove this, it is sufficient to consider weak gravity since a physical solution must be compatible with Einstein's [2] notion of weak gravity (i.e., if there were a dynamic solution for a field equation, it should have a dynamic solution for a related weak gravity [11]). To calculate the radiation, consider further,
G(( ( G(1)(( + G(2)(( , where G(1)(( = (c(c(( + H(1)((, (2b)
H(1)(( ( -(c((((c + (((c( + ((((c(dcd , and ?(((? << 1. (2c)
G(2)(( is at least of second order in terms of the metric elements. For an isolated system located near the origin of the space coordinate system, G(2)(t at large r (= (x2 + y2 + z2 (1/2) is of O(K2/r2) (5,8,17(.
One may obtain some general characteristics of a dynamic solution for an isolated system as follows:
1) The characteristics of some physical quantities of an isolated system:
For an isolated system consisting of particles with typical mass, typical separation , and typical velocities , Weinberg (8( estimated, the power radiated at a frequency ( of order /will be of order
P " ((/)624 or P "8/,
since (/is of order 2. The typical deceleration rad of particles in the system owing this energy loss is given by the power P divided by the momentum, or rad "7/. This may be compared with the accelerations computed in Newtonian mechanics, which are of order 2/, and with the post-Newtonian correction of 4/. Since radiation reaction is smaller than the post-Newtonian effects by a factor 3, if (( c, the velocity of light, the neglect of radiation reaction is perfectly justified. This allows us to consider the motion of a particle in an isolated system as almost periodic.
Consider, for instance, two particles of equal mass m with an almost circular orbit in the x-y plane whose origin is the center of the circle (i.e., the orbit of a particle is a circle if radiation are neglected). Thus, the principle of causality [9,10] implies that the metric g(( is weak and very close to the flat metric at distance far from the source and that g(((x, y, z, t') is an almost periodic function of t' (= t - r/c).
2) The expansion of a bounded dynamic solution g(( for an isolated weak gravitational source:
According (3), a first-order approximation of metric g(((x, y, z, t') is bounded and almost periodic since T(( is. Physically, the equivalence principle requires g(( to be bounded [11], and the principle of causality requires g(( to be almost periodic in time since the motion of a source particle is. Such a metric g(( is asymptotically flat for a large distance r, and the expansion of a bounded dynamic solution is:
g(((nx, ny, nz, r, t') = ((( +(((k)(nx, ny, nz, t')/rk, where n( = x(/r. (4)
3) The non-existence of dynamic solutions:
It follows expansion (4) that the non-zero time average of G(1)(t would be of O(1/r3) due to
((n( = (((( + n( n()/r, (5)
since the term of O(1/r2), being a sum of derivatives with respect to t', can have a zero time-average. If G(2)(t is of O(K2/r2) and has a nonzero time-average, consistency can be achieved only if another term of time-average O(K2/r2) at vacuum be added to the source of (1). Note that there is no plane-wave solution for (1') [9,18].
It will be shown by contradiction that there is no dynamic solution for (1) with a massive source. Let us define
((( = ((1)(( + ((2)(( ; (i)(( = ((i)(( - ((( (((i)cd (cd), where i = 1, 2 ;
and
(((((1)(( = - K T(m)(( . (6)
Then (1)(( is of a first-order; and ((2)(( is finite. On the other hand, from (1), one has
(((((2)(( + H(1)(( + G(2)(( = 0 . (7)
Note that, for a dynamic case, equation (7) may not be satisfied. If (6) is a first-order approximation, G(2)(( has a nonzero time-average of O(K2/r2) (8(; and thus (2)(( cannot have a solution.
However, if (2)(( is also of the first-order of K, one cannot estimate G(2)(( by assuming that (1)(( provides a first-order approximation. For example, (6) does not provide the first approximation for the static Schwarzschild solution, although it can be transformed to a form such that (6) provides a first-order approximation [11(. According to (7), (2)(( will be a second order term if the sum H(1)(( is of second order. From (2c), this would require (((( being of second order. For weak gravity, it is known that a coordinate transformation would turn (((( to a second order term (can be zero) (8,14,17(. (Eq. [7] implies that (c(c(2)(( - (c((((c + (((c( would be of second order) Thus, it is always possible to turn (6) to become an equation for a first-order approximation for weak gravity.
From the viewpoint of physics, since it has been proven that (3) necessarily gives a first-order approximation [11], a failure of such a coordinate transformation means only that such a solution is not valid in physics. Moreover, for the dynamic of massive matter, experiment [16] supports the fact that Maxwell-Newton Approximation (3) is related to a dynamic solution of weak gravity [13]. Otherwise, not only is Einstein's radiation formula not valid, but the theoretical framework of general relativity, including the notion of the plane-wave as an idealization, should be re-examined (see Section 3). In other words, theoretical considerations in physics as well as experiments eliminate other unverified speculations thought to be possible since 1957.
As shown, the difficulty comes from the assumption of boundedness (Section 3), which allows the existence of a bounded first-order approximation, which in turn implies that a time-average of the radiative part of G(2)(( is non-zero (7(. The present method has an advantage over Fock's approach to obtaining logarithmic divergence [13,14( for being simple and clear.
In short, according to Einstein's radiation formula, a time average of G(2)(t is non-zero and of O(K2/r2) [13(. Although (3) implies G(1)(t is of order K2, its terms of O(1/r2) can have a zero time average because G(1)(t is linear on the metric elements. Thus, (1') cannot be satisfied. Nevertheless, a static metric can satisfy (1), since both G(1)(( and G(2)(( are of O(K2/r4) in vacuum. Thus, that a gravitational wave carries energy-momentum does not follow from the fact that G(2)(( can be identified with a gravitational energy-stress (8,17(. Just as G(( , G(2)(( should be considered only as a geometric part. Note that G(t = -KT(m)(t are constraints on the initial data.
In conclusion, in disagreement with the physical requirement, assuming the existence of dynamic solutions of weak gravity for (1) [14,15,19-24( is invalid. This means that the calculations [25,26( on the binary pulsar experiments should, in principle, be re-addressed [12(. This explains also that an attempt by Christodoulou and Klainerman [26( to construct bounded "dynamic" solutions for G(( = 0 fails to relate to a dynamic source and to be compatible with (3) [28] although their solutions do not imply that a gravitational wave carries energy-momentum.
For a problem such as scattering, although the motion of the particles is not periodic, the problem remains. This will be explained (see Section 4) in terms of the 1995 update of the Einstein equation, due to the necessary existence of gravitational energy-momentum tensor term with an antigravity coupling in the source. To establish the 1995 update equation, the supports of binary pulsar experiments for (3) are needed [13].
3. Gravitational Radiations, Boundedness of Plane-Waves, and the Maxwell-Newton Approximation
An additional piece of evidence is that there is no plane-wave solution for (1). A plane-wave is a spatial-local idealization of a weak wave from a distant source. The plane-wave propagating in the z-direction is a physical model although its total energy is infinite [8,10]. According to (3), one can substitute (t - R) with (t - z) and the other dependence on r can be neglected because r is very large. This results in(((xi, t) becoming a bounded periodic function of (t - z). Since the Maxwell-Newton Approximation provides the first-order, the exact plane-wave as an idealization is a bounded periodic function. Since the dependence of 1/r is neglected, one considers essentially terms of O(1/r2) in G(2)((. In fact, the non-existence of bounded plane-wave for G(( = 0, was proven directly in 1991 [9,18].
In short, Einstein & Rosen [4,29] is essentially right, i.e., there are no wave solutions for R(( = 0. The fact that the existing "wave" solutions are unbounded also confirms the nonexistence of dynamic solutions. The failure to extend from the linearized behavior of the radiation is due to the fact that there is no bounded physical wave solution for (1) and thus this failure is independent of the method used.
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].
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