CYLo-Eins Hooft Causality

Bulletin of Pure and Applied Sciences.Vol.27D(No.2)2008:149 -170

THE PRINCIPLE OF CAUSALITY AND THE CYLINDRICALLY SYMMETRIC METRIC OF EINSTEIN AND ROSEN

C. Y. Lo

Applied and Pure Research Institute, 7 Taggart Drive, Unit E, Nashua, NH 03060 USA

ABSTRACT

From the Hulse-Taylor binary pulsars experiment, it has been found that the nonexistence of gravitational wave solutions is due to a violation of the principle of causality. Nevertheless, some still believe that the wave solutions exist because singularities in the solution of Einstein & Rosen are removable. It is shown that the violation of causal-ity can be proven from the metric form of Einstein-Rosen type, in addi-tion to explicit cylindrical “waves”. Metrics of Einstein-Rosen type vio-late Einstein’s equivalence principle, and violate the principle of causal-ity due to the impossibility of having valid sources. Concurrently, the cylindrical symmetry metric, obtained by G. ‘t Hooft from combining “waves”, is also discussed. There are two fundamental errors, namely: 1) the plane wave has been implicitly extended beyond its physical valid-ity as an idealization, and 2) the integration over the angle cannot be justified with a physical process.

Key words: Einstein’s equivalence principle, covariance principle, Euclidean-like structure, prin ciple of causality, plane-wave. 04.20.-q, 04.20.Cv

“Science sets itself apart from other paths to truth by recognizing that even its greatest practitioners sometimes err.” -- S. Weinberg, Physics Today, November 2005.

1. INTRODUCTION

A common mistake among theorists (Lo 2000b) including Einstein (Ein-stein et al 1938), Feynman (1996), Landau & Lifshitz (1962), etc., was assuming the existence of dynamic solutions for the Einstein equation of 1915. This issue of dynamic solutions was raised by Gullstrand (1921) in his report to the Nobel Committee. Due to conceptual errors such as ambiguity of coordinates as pointed out by Whitehead (1922), Fock (1964), and Zhou (1983), many cannot reconcile the non-existence of dynamic solutions with the three accurate predic-tions. It was not until 1995 that based on the principle of causality, the nonexis-tence of dynamic solutions is proven (Lo 1995, 1999a) and related issues are ad-dressed subsequently (Lo 2000b). This becomes possible (Lo 2003b) because it is proven that a physical space must have a frame of reference that has a Euclidean-

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like structure. 1) Meanwhile, unphysical solutions were accepted (Kramer et al. 1980) because Einstein’s equivalence principle was not well understood (see Ap-pendix A).

To explain the Hulse-Taylor binary pulsars experiment, it is necessary to modify the Einstein equation with an added source term, the gravitation energy-stress tensor (with an antigravity coupling), to accommodate the waves (Lo 1995, 1999a) and to have a valid linearized equation. Since the gravitational wave, whose sources are energy-stress tensors, carries energy-momentum, a gravita-tional wave should carry a source along. Then, it becomes clear (Lo 2000b) that the non-existence of gravitational waves is due to a violation of the principle of causality (see Section 2).

Historically, Einstein & Rosen (1937) could be considered as the first to discover the non-existence of wave solutions, but editors of the Physical Review found that the singularities they discovered are removable (Kennefick 2004). This led to a self-deceptive satisfaction that subsequently hindered progress in phys-ics. Also Christodoulou & Klainerman (1993) claimed to have constructed dy-namic solutions for the Einstein equation because they made some basic errors in mathematics (Lo 2000a). Moreover, the editors of the Physical Review and other journals failed to identify the violation of physical principles such as the princi-ple of causality (Lo 2000b, 2003a, 2007c).

Surprisingly some still clinch on the cylindrical solution of Einstein-Rosen type. The reason seems to be that this provides a seemingly close mathe-matical analogy with the case of electromagnetism. For instance, ‘t Hooft pro-vided an example (Lo 2006b) to illustrate his claim of the validity of the Einstein equation for the dynamic case. His goal is to justify the linearization of the Ein-stein equation although this has been proven invalid for the dynamic case (Lo 1995, 2000b).

In this paper, it will be shown that the cylindrical solution of Einstein-Rosen type is invalid because both Einstein’s equivalence principle and the prin-ciple of causality are violated. To illustrate these, the example of ‘t Hooft is ana-lyzed together the solution of Weber and Wheeler (1957). The violation of these principles will be identified and discussed with the details of the examples. It is hoped that thereafter theorists would study the problem of gravitation anew with adequate physical considerations included.

2. THE PRINCIPLE OF CAUSALITY

The time-tested assumption that phenomena can be explained in terms of identifiable causes is called the principle of causality (Lo 1995, 2000b). This prin-ciple is the basis of relevance for all scientific investigations. This principle is commonly used in symmetry considerations in electrodynamics.

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The principle of causality and the cylindrical symmetry metric of Einstein and Rosen

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In general relativity, Einstein and subsequent theorists have used this

principle implicitly on symmetry considerations (Lo 2000b) such as for a circle in a uniformly rotating disk and the metric for a spherically symmetric mass distri-bution. In fact, a crucial argument in the proof of the nonexistence of dynamic solutions of Einstein equation (Lo 1995, 2000b) is based on symmetry considera-tions due to the principle of causality.

However, physicists have not considered its other consequences until re-cently. For instance, parameters unrelated to any physical cause in a solution are not allowed and a dynamic solution must be related to an appropriate dynamic source. Moreover, that the weak sources would produce weak gravity is, in fact, the theoretical foundation of Einstein’s requirement on weak gravity. Neverthe-less, this principle is often neglected in the consideration of solutions (Lo 2000b; Kramer et al.1980; Einstein & Rosen 1937).

A reason is that since the coordinates are ambiguous, it is often difficult to apply the principle of causality in a logical manner other than implicitly. The negligence is apparently habit forming, and thus theorists who are still dominat-ing the field of general relativity essentially forgot this principle. The so-called “covariance principle” that many theorists incorrectly believed 2) is also respon-sible for such a strange situation. (Besides, few would venture out to related fields in mathematics and physics. 3))

Thus, it would be necessary to clarify the physical meaning of coordi-nates. Then Einstein’s requirement on weak gravity can be justified (Lo 1999b). On the other hand, if the physical meaning of space coordinates is not clear, it is difficult to justify in applying this principle to symmetry considerations as Ein-stein and others implicitly did.

An illustrative example for a violation of the principle of causality is the metric of Bondi, Pirani & Robinson (1959), claimed as a wave from a distant source,

ds2 = exp(2φ)(dτ2 – dξ2) – u2[cosh2β (dη2 + dζ2) + sinh2β cos2θ (dη2 – dζ2)

– 2sinh2β sin2θ dηdζ], (1)

where φ, β, θ are functions of u (= τ – ξ ). It satisfies the equation (i.e., their eq. [2.8]),

2φ' = u(β' 2 + θ' 2 sinh2 2β). (2)

Metric (1) is unbounded, although its frame (ξ, η, ζ) has a Euclidean-like struc-ture. Thus they had to claim that Einstein’s requirement for weak gravity is meaningless.

However, such a rejection is insufficient to justify metric (1). When grav-ity is absent, i.e. φ = β = 0, the metric is reduced to


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ds2 = (dτ2 – dξ2) – u2 (dη2 + dζ2). (3)

On the other hand, the metric for no gravity should be the flat metric or its equivalence as in the case of “cylindrical waves” of Einstein & Rosen (1937). Thus, to justify metric (1), one must show that, for the given frame, (3) could be equivalent to the flat metric. Moreover, metric (1) violates the principle of causal-ity since there is no parameter of physical causes to be adjusted such that metric (1) becomes equivalent to the flat metric.

Consider another metric accepted as valid by Penrose (1964) as follows:

ds2 = du dv + H du2 – dxi dxi, where H = hij(u)xixj, (4)

and

hii(u) ≥ 0, hij = hji , where u = ct - z, and v = ct + z . (5)

Here t is the time coordinate; and x, y, z are the space coordinates; and hij(u) is an energy-stress tensor related to the cause of this gravity, an electromagnetic plane-wave.

Metric (4) is a Lorentz manifold. However, independent of hij(u), H can always be arbitrarily large. This is incompatible with Einstein's notion of weak gravity and the correspondence principle. Moreover, in general H = hij(u)(xi – ai)(xj – aj) and the parameters are chosen as (a1, a2) = (0, 0). Thus, the principle of causality is violated since parameters ai (i = 1, 2) are unrelated to any physical cause.

3. THE CYLINDRICAL SYMMETRY METRICS OF EINSTEIN AND ROSEN

Let us examine their cylindrical “waves” of Einstein & Rosen (1937) again. In coordinates, ρ, ϕ, and z, their solution is

ds2 = exp(2γ – 2Ψ)(dT2 – dρ2) – ρ2exp(–2Ψ)dϕ2 – exp(2Ψ)dz2 (6)

where T is the product of the velocity of light and the time coordinate. Its frame of reference has the Euclidean-like structure. γ and ψ are functions of ρ and T. They satisfy

Ψρρ + (1/ρ)Ψρ – ΨTT = 0, (7a)

γρ = ρ[Ψρ2 + ΨT

2], and γ T = 2ρΨρΨ T. (7b)

Also, the function γ satisfies an inhomogeneous linear equation of Maxwell-type,

γρρ + γρ/ρ – γTT = 2ψT2 (8)

When gravity is absent (i.e., γ = ψ = 0), the reduced metric is equivalent to the flat metric.



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Weber & Wheeler (1957) claimed, “We concluded that many of the oth-

erwise apparently paradoxical properties of this cylindrical wave can be under-stood by taking into account the analogy between gravitational waves and elec-tromagnetic waves, and the special demands of the equivalence principle, which rules out a special role for any particular frame of reference.” 4) Their claims could be valid if γ(r, T) had a physical solution. However, their own solution im-plies otherwise.

Weber and Wheeler (1957) obtained a solution as follows:

　Ψ = AJ0 (ωρ) cos ωt +BN0 (ωρ) sinωt, (9a)

where A and B are constants. The second function γ in the special case B = A, re-duces to

γ = 21

A2ωρ{J0(ωρ)J0’(ωρ) + N0(ωρ)N0’(ωρ) + ωρ[(J0(ωρ))2 + (J0’(ωρ))2

+ (N0(ωρ))2 + (N0’(ωρ))2] + [J0(ωρ)J0’(ωρ) – N0(ωρ)N0’(ωρ)] cos2ωT

+ [J0(ωρ)N0’(ωρ) + N0(ωρ)J0’(ωρ)] sin2ωT} – π2

A2ωT (9b)

According to (9b), exp(2γ) gets very large as T gets large negatively. (Note that T and ωρ should have been respectively t and ωρ/c.) Moreover, since exp(2γ) ap-proaches zero as T gets large positively, the condition for weak gravity (1 >> |γμν|) would fail.

Moreover, metric (6) cannot satisfy coordinate relativistic causality. We-ber and Wheeler agreed with Fierz's analysis that γ is strictly positive where Ψ(ρ, T) ≈ 0 for large ρ (1957). On the other hand, for a cylindrical coordinate system, Einstein’s equivalence principle implies time dilation and space contractions, and this would mean

– gρρ ≥ 1 ≥ gtt , – gϕϕ/ρ2 ≥ 1 , and – gzz ≥ 1 . (10a)

These would imply coordinate relativistic causality (Lo 2002). Thus, from metric (6) one has exp (2γ) ≤ 1 and exp (2γ) ≤ exp(2Ψ). It thus follows that γ ≤ 0. Hence, the condition γ > 0 cannot be met. In fact, it follows (10a) directly that

(2γ – 2Ψ) ≥ 0 ≥ (2γ – 2Ψ), –2Ψ ≥ 0 , and 2Ψ ≥ 0. (10b)

Thus Ψ = γ = 0. This shows that there is no physical wave solution for Gμν = 0.

This above analysis shows not only there is no wave solution, but also no physical solution of any kind. Unlike the case of plane waves, a cylindrically symmetric metric must have a source because it is not a local idealization of waves from distant sources. It will be shown in the next section that it is impos-sible to have appropriate sources.


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4. REMARKS ON THE MAXWELL-NEWTON APPROXIMATION

To the above end, some review of the Maxwell-Newton Approximation would be useful. It was incorrectly believed that the linear Maxwell-Newton Ap-proximation always provides the first order approximation for the Einstein equa-tion (Einstein 1954; Wald 1984; Will 1981). This belief was verified for the static case only. For a dynamic case, however, this is no longer valid (Lo 1995, 2000b). Nevertheless, some theorists believe that the cylindrically symmetric metric could be an exception.

To discuss this, let us consider the Maxwell-Newton Approximation as fol-lows:

21

∂ c∂c γ μν = – K T(m) μν , where γ μν = γμν – 21

ημν(ηcdγcd) (11a)

and

γ μν(xi, t) = – π2K

∫ r1

Tμν[yi, (t – r)]d3y, where r2 =∑=

−3

1

2)(i

ii yx . (11b)

From (11b), a source would give γμν as of the first order. From (11a), it is clear that the linearized harmonic gauge ∂ c γ cd = 0 would imply the conservation law, ∂ cT(m)cd = 0. In turn, as pointed out by Wald (1984), this would mean that the test bodies move on geodesics of the flat metric ημν. On the other hand, the covariant conservation law,

∇cT(m)cd = 0, (12)

implies only that ∂ c γ cd is of the second order. Thus, theoretical consistency re-quires that the linearized harmonic gauge is not exact. In other words, for gen-eral relativity, a tensor T(m)cd that satisfies ∂ cT(m)cd = 0 is invalid.

Consider the metric of Einstein & Rosen (1937). For the case of weak grav-ity, we have the lowest order:

γ tt(ρ, t) = –2Ψ2, γ zz(ρ, t) = – 4Ψ(ρ, t)+2[γ(ρ, t)+ Ψ2], γ xy(ρ, t) = –2γ(ρ, t) xy/ρ2,

γ xx (ρ, t) = 2γ(ρ, t) y2/ρ2+2Ψ2, and γ yy (ρ, t) = 2γ(ρ, t) x2/ρ2+2Ψ2. (13)

Thus,

∂ c γ ct = – 4Ψ(r, t )Ψt(r, t), ∂ c γ cz = 0, (14a)

∂c γ cx = 2γ(ρ, t)[x/ρ2] – 4Ψ(r, t )Ψx(r, t), (14b)

and

∂c γ cy = 2γ(ρ, t)[y/ρ2] – 4Ψ(r, t )Ψy(r, t). (14c)

Thus, the linearized harmonic gauge would be satisfied if γ(ρ, t) is bounded and



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|γ(ρ, t)| ≈ O(ρ), as ρ → 0. (14d)

However, in the solution of Weber & Wheeler, γ (r, t) is not bounded.

Even if γ is bounded and (14d) is satisfied, since only γ z z could be of the first order, the gravity should be generated only from T(m) zz . Thus masses are moving in the z direction. Since γ zz (ρ, t) is independent of z, the motion is uni-form with respect to z. Moreover, since γ tt (ρ, t) is of second order, there is no T t t to generate the metric. Thus, there is no valid source for a cylindrically symmet-ric metric of Einstein-Rosen type.

In other words, the principle of causality is violated. The metric (6) being in violation of Einstein’s equivalence principle is also a statement of the fact that it is impossible to move an infinitely long mass uniformly in the axial direction, but there is actually no mass involved. The only hope seems to have waves that could be argued as having the sources at infinity. This is probably the motivation of ‘t Hooft’s metric solution.

5. THE CYLINDRICAL SYMMETRY ‘WAVE’ SOLUTION OF ‘t Hooft

The cylindrical metric constructed by ‘t Hooft (Lo 2006b) provides an in-teresting example violating the principle of causality. His original goal is, how-ever, to show the existence of wave solutions. His cylindrical metric is as follows:

2)cos(2

0),( ϕαπ

ϕ rtedAtr −−∫=Ψ , (15)

where A and α (> 0) are free parameters. For simplicity, take them to be one. |ψ| is everywhere bounded. He claimed that, at large values for t and r, the stationary points of the cosine dominate, so that there are peaks at r = |t|. And this is a packet coming from r = ∞ at t = –∞ bouncing against the origin at t ≅ 0 (always obeying the correct boundary condition there), and moving to r = ∞ again at t → ∞. The metric satisfies equations:

01=Ψ−Ψ+Ψ ttrrr r , )( 22

trr r Ψ+Ψ=γ , trt r ΨΨ= 2γ . (16)

At |t| >> r, the function ψ drops off exponentially, and at r >> |t| as a power:

222tr −

→Ψπ

(17)

‘t Hooft claims that his metric solution is bounded, and has a weak limit.

However, it should be noted that Ψ(r, t) is not a wave packet as he claimed. Let us calculate Ψr and Ψt as follows:


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)cos(22)cos(2

0ϕϕ ϕπ

rted rtt −−=Ψ −−∫ ))(coscos(2

2)cos(2

0ϕϕϕ ϕπ

rted rtr −=Ψ −−∫ .(18)

Consider the case t = 0, and we have 2)cos(2

0)0,( ϕαπ

ϕ redr −∫=Ψ

0cos22)cos(2

0=−=Ψ −∫ ϕϕ ϕπ

red rt , but ϕϕ ϕπ 22)cos(2

0cos2 red r

r−∫=Ψ ≠ 0. (19a)

Consider another case r = 0, and we have 2)(2

0),0( tedt −∫=Ψ ϕ

π

))(2(2)(2

0ted t

t −=Ψ −∫ ϕπ

≠ 0, but )cos)()(2(2)(2

0ϕϕ

π−−=Ψ −∫ ted t

r = 0. (19b)

Thus, |Ψr(r, t)| ≠ |Ψt (r, t)|, and their ratio changes with (r, t). Thus, Ψ (r, t) is not a plane wave packet, and γ(r, t) might have a bounded solution. However, since

2)( trtr r Ψ±Ψ=± γγ , (20)

the function γ(r, t) has the same kind of problem (see also next section).

One might argue that since (15) satisfies a Maxwell-type equation in vac-uum, its wave should have the same phase speed. However, such an argument is valid only for a physical wave packet. Since (15) does not satisfy Einstein’s equivalence principle, other kinds of violation are expected as shown in the solu-tion (4) of Weber and Wheeler (1957).

‘t Hooft claimed, “The other function γ(r, t) is found by integrating one of the two other equations (they of course yield the same value).” γ drops off expo-nentially at | t | >> r and for r >> | t | it goes asymptotically to:

.)(

222

2Cst

trr

+−

−→

πγ (21)

Note the t ↔ –t symmetry throughout. This estimation of ‘t Hooft (2007) is based on asymptotic expansion as follows:

　Ψ (r, t) = A ⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

++ )1(

16)4123(3

2212

75

42

3

2

rO

rtt

rt

rπ (22)

for r >> t. This gives for the function the asymptotic expansion:

　γ (r, t) = ))1(4

244892

832(86

42

4

2

22

rO

rtt

rt

rCA +

++−

+−−π (23)

An implicit assumption of this approach is that the expansion (23) converges. Moreover, at r→0, the expansion is



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　γ (r, t) = A2 π )()21(1 4222

rOtre t +⎥⎦⎤

⎢⎣⎡ −+−

. (24)

Then, using the same implicit assumption as above, for r →0 one has

　γ (r, t) = )(8 422222 2

rOertA t +−π . (25)

Then, (21) was derived and he concludes that

4πA2 > γ(r, t) > 0. (26)

As ‘t Hooft requested, details are included in Appendix B. Note also that just as in the solution (9) of Weber and Wheeler (1957), there are problems related to γ according to eq. (20). Nevertheless, these mathematical details are no longer im-portant since it has been proven that this type of metric is not physically valid.

The plane-wave of the Maxwell equation is a local idealization with no sources. However, for a cylindrical solution, it is different. Understandably, ‘t Hooft concluded that (16) should have the following extended form,

),(1 trJr rrrtt =Ψ−Ψ−Ψ where J(r, t) = δ(t) Φ2(r) + δ’(t)Φ1(r) , (27a)

which requires

　Ψ (r, t) → Φ 1(r) , 　Ψ t(r, t) → Φ2(r) , as t ↓ 0 (27b)

Note that Φ2(r) = 0 in the given example. However, having a delta function of t, the source in (27) is invalid, because this solution cannot be attributed to the mo-tion of accelerated masses. Thus, the principle of causality is violated. Note that the causality of ‘t Hooft is based on a Maxwell-type equation, which is also satis-fied by metric (9).

Thus, a major issue would be whether (15) is a wave packet because it was hoped that one could argue that the sources were at infinity. However, (15) is not a wave packet that requires all its components to have the same wave speed. Moreover, there is no source related to r = ∞, and (15) is invalid (see also next section). In conclusion, the principle of causality is violated since there is no ap-propriate source for packet (15).

6. PHYSICS RELATED TO THE PLANE WAVE AS AN IDEALIZATION

To see the physics clearly in a simple manner, the method of idealization is often used. A good example is the plane wave. A plane wave with frequency


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ω, uω (x, t) is an idealization of a wave in a ray with a single frequency from a dis-tant source. In such an idealization of a section of the ray, the change of ampli-tude over distance is neglected and thus the source would not be seen directly, but it is implicitly included in the amplitude. Thus, although the total energy of a plane wave is infinite, the energy density of a plane wave is finite. Thus, a plane wave is not a real wave, but it is physically valid as an idealization of a section of wave within a narrow ray of propagation.

In a plane wave idealization, the direction of propagation is essential. The characteristic of plane wave is used to form wave packets in terms of linear combinations. This is more realistic since a real wave generally contains multiple frequencies, and its amplitude is noticeably non-zero only in a finite range. Then, a wave packet is

u(x, t) = ωωπ

ω dtxuA ),()(21

∫∞

∞− (28)

Note that the coefficients A(ω) could be arbitrary. However, the propagation di-rection and speed are maintained since

∂ x uω(x, t) = –∂ t uω(x, t) implies ∂ x u(x, t) = –∂ t u(x, t) (29)

Thus, such a wave packet is a valid idealization. An example of the wave packet is

2)(),( xteAtxu −−= (30)

It is understood that the validity of (30) is within the neighborhood of a narrow ray propagating in the x-direction.

Now, consider that x = r cos ϕ in a coordinate system (r, ϕ, z). Then, ‘t Hooft (2007) claimed that his solution,

2)cos(2

0),( ϕαπ

ϕ rtedAtr −−∫=Ψ (31)

is obtained by superimposing plane wave packets of the form exp [−α(x − t)2] ro-tating them along the z axis over angle ϕ, so as to obtain a cylindrical solution. Note that since the integrand exp[−α(t − r cosϕ)2] = exp[−α(t − x)2], there is no rota-tion along the z axis. The function exp[−α (t − x)2] is propagating from x = – ∞ to x = ∞ as time t increases.

However, (28) is integrated over a parameter ω unrelated to the x-axis; whereas the “superimposition” (31) is integrated over ϕ(x, y). Since, (31) is a combination that involves the coordinate ϕ (x, y), it is not a superimposition of plane waves propagating along the x-axis. Furthermore, the integration over all angles ϕ is a problem that would violate the requirement of the idealization be-



159

cause it requires that the plane wave is valid over the whole x-y plane. Thus, function (31) is not valid as an idealization in physics.

Alternatively, (31) could be interpreted as the sum of infinitely many “wave packets”, each of which is in the narrow ray at angle ϕ with a different amplitude at a point (r, t) and with a different phase speed,

ϕcos1

=tr

. (32)

Hence, the wave speed would be ranging from –∞ to –1 and 1 to +∞ at different rays. This is also unacceptable in physics since energy should not propagate faster than c. Note that since the wave packet exp[–α(t – x)2]dy is propagating along x-axis, physically it is incompatible with the “packet” exp[–α(t – r cosϕ)2]dϕ propagating along r.

Therefore, in solution (31), two fundamental errors have been made, namely: 1) the plane wave has been implicitly extended beyond its physical va-lidity, and 2) the integration over dϕ is a process without a valid physical justifi-cation.5) These are other forms of violation of the principle of causality.

7. DISCUSSION AND CONCLUSIONS

Einstein’s equivalence principle is important, thus it is necessary to clar-ify the fine points in its application. Moreover, the application of the principle of causality could be subtle at times. To illustrate this, examples of the violation of these basic principles have been given. The constructed solution of ‘t Hooft pro-vides an additionally very good example to illustrate the power of these princi-ples. A basic difference between a plane-wave metric and a cylindrical symmetry metric is that the latter must have a source.

Weber & Wheeler (1957) and ‘t Hooft (Lo 2006b) drew physical interpre-tations from Ψ, which satisfies a Maxwell-type equation. However, γ would have a “wave” source as shown by eq. (3). Mathematically a plane-wave source would guarantee the solution of γ to be invalid (Lo, Chan, & Hui 2002). (How-ever, Ψ in the solution of ‘t Hooft is not a wave.) Physically, a violation of the principle of causality should be expected since this metric does not satisfy Ein-stein’s equivalence principle.

In ‘t Hooft’s solution, components of Ψ (r, t) would propagate at various speeds larger than the light speed. His metric further illustrates the absence of a valid physical source just as Einstein’s equivalence principle would imply. Moreover, Ψ (r, t) of ‘t Hooft is also physically invalid even if considered for the


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case of Maxwell’s equations (see Section 6). In short, Ψ is constructed with opera-tions that are invalid in physics.

Thus, a violation of causality is expected and it supports the physical va-lidity of Einstein’s equivalence principle. Nevertheless, some theorists still do not know what Einstein’s equivalence principle is. For instance, having mistaken it the same as Pauli’s version, Synge (1971) incorrectly claimed Einstein’s equiva-lence principle as nonessential. Similarly, Wald (1984) simply ignores it (see also Appendix A).

‘t Hooft’s metric further illustrates the absence of a physical source just as Einstein’s equivalence principle would imply. For the gravitational waves that VIRGO and LISA want to detect, the sources of the plane waves are not dis-cussed. However, it is known that such plane waves are solutions of a modified Einstein equation (Lo 1995, 2000b), but are beyond the 1915 Einstein equation (Lo 2006a).

A common error 6) was to assume the existence of dynamic solutions for the Einstein equation (Lo 2000b). However, the non-existence of dynamic solu-tions was clarified because the principle of causality can be applied (Lo 2000b) after the physical meaning of coordinates is clarified with Einstein’s equivalence principle. Nevertheless, many still mistake (Lo 2007b) Einstein’s equivalence principle to be the same as Pauli’s (1958) version; and the formula of Landau & Lifshitz for space contractions was rejected (Lo 2005b). In addition, many still in-correctly regard Einstein’s equivalence principle of 1921 to be the same as the 1911 assumption of equivalence (Cheng 2005).

In general relativity, problems are often related to inadequate under-standing of Einstein’s equivalence principle. Sometimes even Einstein himself also did not understand his own principle well. Otherwise, he would not have proposed a metric form that has its violation. However, this is no longer a sur-prise since it has been shown that Einstein’s theory of measurement is inconsis-tent with his equivalence principle (Lo 2003b, 2005). Many theorists also do not understand Einstein’s equivalence principle (Lo 2007b), except possibly a few such as Zhou Pei-Yuan (1983, 1987). 7) 8)

In addition, Liu & Zhou (1985) recognized that physical requirements must be considered for a solution of plan-waves. In contrast, Penrose (1964) ac-cepted solutions with unphysical parameters (Lo 2000b). Unaware of a violation of the principle of causality, Christodoulou & Klainerman (1993) claimed to have dynamic solutions constructed, because they made errors in mathematics (Lo 2000a).9) On the other hand, ‘t Hooft’s error is essentially in the area of physical understanding. In fact, quite a few theorists do not understand the principle of causality adequately (Lo 2007c). It is hoped that the scientific community will give the necessary help to those relativists.



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The cylindrical symmetry "wave" solution of Einstein and Rosen is inva-

lid since both Einstein’s equivalence principle and the principle of causality are violated. In fact, some fail to understand the principle of causality because they do not understand Einstein’s equivalence principle. The principle of causality is the oldest and simplest principle in physics. The inadequate deliberation of this principle and/or the difference between mathematics and physics are probably just due to carelessness. 10) However, this is not an isolated problem even in the Royal Society.

It is hoped that this paper will fairly settle the historical account of the events between Einstein and the Physical Review. More important, theorists will no longer ignore physical principles in their work on general relativity. Remem-ber that the serious error of accepting Einstein’s interim covariance principle was made by almost the whole physics community. Nevertheless, the cause of such an error is merely due to inadequate deliberation of the difference between phys-ics and mathematics (Lo 2008a).

Just like Einstein, ‘t Hooft also made mistakes. However, this does not di-minish their status as great physicists earned by their contributions. In fact, new discoveries just further reaffirm Einstein’s great contribution since implications of general relativity have far reaching consequences such as unification that only recently we begin to appreciate (Lo 2007a, 2008b).

ACKNOWLEDGMENTS

This paper is dedicated to Heng for unfailing encouragements over years. The author is grateful for stimulating discussions with S. L. Cao, David P. Chan, A. J. Coleman, S. -J. Chang, Richard C. Y. Hui, Liu Liao, and A. Napier on plane-waves and the analysis of Weber and Wheeler on the cylindrical symmetry met-ric of Einstein and Rosen. Special thanks are to G. ‘t Hooft and John Dyson, Uni-versity of Leeds UK, for extensive discussions on the solution of ‘t Hooft; and A. Napier for editing the whole manuscript. This work is supported in part by In-notec Design, Inc., U.S.A.

Appendix A: Einstein’s Principle of Equivalence, the Einstein-Minkowski Condition

As shown by Einstein (1954; Einstein et al. 1923), a consequence of his equivalence principle is the Einstein-Minkowski condition that the local space of a particle under gravity must be locally Minkowskian, from which he obtained the time dilation and space contractions. However, many others often regarded this condition as non-essential (Norton 1989). Although Einstein used it in his initial paper and his book, “The Meaning of Relativity”, many may still have missed this important point (Lo 2007b).


C. Y. Lo

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First, let us state Einstein’s principle, which is based on the equivalence of inertial mass and gravitational mass due to Galileo, with Einstein’s own words. In his book, “The Meaning of Relativity”, Einstein (1954) wrote:

‘Let now K be an inertial system. Masses which are sufficiently far from each other and from other bodies are then, with respect to K, free from acceleration. We shall also refer these masses to a system of co-ordinates K’, uniformly accelerated with respect to K. Relatively to K’ all the masses have equal and parallel accelerations; with respect to K’ they behave just as if a gravitational field were present and K’ were unaccelerated. Overlooking for the present the question as to the “cause’ of such a gravitational field, which will occupy us latter, there is nothing to prevent our conceiving this gravitational field as real, that is, the conception that K’; is “at rest” and a gravitational field is present we may consider as equivalent to the conception that only K is an ”al-lowable” system of co-ordinates and no gravitational field is present. The assumption of the complete physical equivalence of the systems of coordinates, K and K’, we call the “principle of equivalence;” this prin-ciple is evidently intimately connected with the law of the equality be-tween the inert and the gravitational mass, and signifies an extension of the principle of relativity to coordinate systems which are non-uniform motion relatively to each other.’

Later, Einstein made clear that a gravitational field is generated from a space-time metric. Thus, his principle was proposed for the gravity as an integral part of the physical space.

Einstein’s equivalence principle is different from Einstein’s 1911 pre-liminary assumption on the equivalence between uniform acceleration and uni-form Newtonian gravity (Einstein et al. 1923). Fock (1964) found it impossible to show that the related metric is of the following form, ds2 = g t t (x) c2dt2 – dx2 – dy2 – dz2 (Lo 2007b).

What is new in Einstein’s equivalence principle in 1916 is the claim of the Einstein-Minkowski condition as a consequence (Einstein et al, 1923, p. 161). While Einstein’s 1911 preliminary assumption was based essentially on his intui-tion, the Einstein-Minkowski condition additionally has its foundation from mathematical theorems (Synge 1971) in Riemannian geometry as follows:

Theorem 1. Given any point P in any Lorentz manifold (whose metric sig-nature is the same as a Minkowski space) there always exist coordinate systems (xμ) in which ∂gμν/∂xλ = 0 at P.

Theorem 2. Given any time-like geodesic curve Γ there always exists a co-ordinate system (so-called Fermi coordinates) (xμ) in which ∂gμν/∂xλ = 0 along Γ.



163

In these theorems, the local space of a particle is locally constant, but not neces-sarily Minkowski. However, after some algebra, a local Minkowski metric exists at any given point and that along any time-like geodesic curve Γ, since a moving local constant metric exists. The only condition is that the space-time metric has a proper Minkowski signature.

What Einstein added to these theorems is that physically such a locally constant metric must be Minkowski. Such a condition is needed for special rela-tivity as a special case (Einstein et al. 1923). In a uniformly accelerated frame, the local space in a free fall is a Minkowski space according to special relativity. It should be noted, however, that validity of the Einstein-Minkowski condition used in Einstein’s calculations (Einstein 1954; Einstein et al. 1923) is assumed only.

Accordingly, Pauli’s version is essentially a simplified but corrupted version of these theorems. Pauli’s (1958) version is as follows:

“For every infinitely small world region (i.e. a world region which is so small that the space- and time-variation of gravity can be neglected in it) there always exists a coordinate system K0 (X1, X2, X3, X4) in which gravitation has no influence either in the motion of particles or any physical process.”

Thus, Pauli regards the equivalence principle as merely the mathematical exis-tence of locally constant spaces, which may not be locally Minkowski. In addition, Pauli invalidly extended the removal of uniform gravity to the removal of grav-ity in general. However, in spite of Einstein’s objection (Norton 1989), his equiva-lence principle was commonly but mistakenly regarded the same as Pauli’s ver-sion.

Einstein’s equivalence principle is also often misinterpreted. For instance, this happens in the highly praised book of Will (1981), which also misinterpreted (Lo 1995, 2000b) the binary pulsars experiment of Hulse & Taylor. Will (1986, p. 20) claimed “’Equivalence’ came from the idea that life in a free falling laboratory was equivalent to life without gravity. It also came from the converse idea that a laboratory in distant empty space that was being accelerated by a rocket was equivalent to one at rest in a gravitational field.” Moreover, the British Encyclo-pedia also stated Einstein’s Equivalence Principle incorrectly and ignored the Einstein-Minkowski condition.

Apparently, Pauli (1958), and Will (1981, 1986), overlooked (or disagreed with) Einstein’s (Einstein et al., p.144) remark, “For it is clear that, e.g., the gravi-tational field generated by a material point in its environment certainly cannot be ‘transformed away’ by any choice of the system of coordinates…” Now, it should be clear that Pauli and his followers knew little about functional analysis in mathematics.


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164

Ignoring the Einstein-Minkowski condition, Misner, Thorne, & Wheeler (1973; p. 386) claimed that Einstein’s equivalence principle is as follows: -

“In any and every local Lorentz frame, anywhere and anytime in the universe, all the (Nongravitational) laws of physics must take on their familiar special-relativistic form. Equivalently, there is no way, by experiments confined to infinitestimally small regions of spacetime, to distinguish one local Lorentz frame in one region of spacetime frame any other local Lorentz frame in the same or any other region.”

In their eq. (40.14), they got an incorrect conclusion on the local time of the earth in the solar system. Ohanian & Ruffini (1994, p. 198) also had the same problems as shown in their eq. (50). However, Eddington (1975), Straumann (1984), Wald (1984), and Weinberg (1972) did not make the same mistake.

Moreover, based on invalid calculations of Fock, Ohanian & Ruffini (1994), and Wheeler also claimed that Einstein’s equivalence principle is inva-lid. Nevertheless, many claimed that his version is a “standard theory” al-though the founders of the International Society on General Relativity and Gravitation have not reached a common consensus. Note also that Herrera, Santos & Skea (2003) publish an incorrect paper on E = mc2 (Lo 2007b), and missed the evidence for unification (2007a).

Appendix B: The Related Calculation Details and Statements of G. ‘t Hooft

Consider the solution 2)cos(2

0),( ϕαπ

ϕ rtedAtr −−∫=Ψ , drrrr

tr )()(0

22∫ Ψ+Ψ=γ (B1)

Here, A can be as large or as small as we please a), and α must be positive. It is clear that γ (r) is a monotonously increasing function of r.

Using Mathematica®, the asymptotic expansion of Ψ(r, t) at r >> |t| is found to be (we can add as many terms as we want):

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++

++=Ψ )1(

16)4123(3

2212),(

752

422

3

2

rO

rtt

rt

rAtr

ααα

αα

απ

(B2)

This gives for the function γ (r, t) the asymptotic expression:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++−

+−−= )1(

4)24489

2832),(

863

422

4

2

22

rO

rtt

rt

rCAtr

ααα

αα

απγ (B3)

This series expansion of Ψ (r, t) at r → 0 is



165

)())21(1(2),( 4222

rOtreAtr t +−+=Ψ − ααπ α (B4)

and γ (r, t) is therefore

)(8),( 422222 2

rOertAtr t += − ααπγ (B5)

As a consistency check, one can verify these expansions obey

trt rtr ΨΨ= 2),(γ (B6)

To estimate the constant C, and find a bound for γ (r, t), we calculate it at t = 0, where

22

2

1

0 1

14)( xrex

dxAr α−

−=Ψ ∫ ; (B7)

Ψt(r)= 0, 01

8)(22

2

21

0<

−−=Ψ −∫ xr

r ex

xdxrAr αα (B8)

By partial integration:

22

)21(18)( 2221

0xr

r exrxdxrAr ααα −−−=Ψ− ∫ . (B9)

Since the second term is negative, this is bounded by

rKxdxrAr 021

018 =−<Ψ ∫α , K0 =2πAα. . 　 (B10)

and also by

11

088

22

KAedxrA xrr ==<Ψ −∫ παα α . (B11)

To find a good bound at large r, we write (B8) as

ϕαπ

πϕα

22 sin22/

2/sin4)( r

r edxrAr −−∫−=Ψ . (B12)

In the integration region, we have

ϕϕ <sin , but πϕϕ /2sin > , (B13) so that

22

/42 /4222

rKedrA rr =<Ψ −∞

∞−∫πϕαϕϕα ,

απ

4

2/7

2AK = . (B14)

The function γ (r) is an increasing function and since γr = Ψr2 at t = 0, we have the bounds

320 rKr <γ , rKr

21<γ , 32

2 rKr <γ . (B15)


C. Y. Lo

166

It is best to use the first and the last one. We find that these two bounds, coming from (B10) and (B14) are equal at

3/1210 )/( KKrr == . (B16)

This gives the bound

3/1623/42

3/20

20

240

200

2

163

43/

21

41)(

2

0

0

πγπ AKKrKrKdrrCAr

r==+<⎟

⎠⎞⎜

⎝⎛ += ∫ ∫

∞;

7496.26163 3/13 =< πC (B17)

Thus, we found an absolute bound on the constant C, which bounds the mo-notonously increasing function γ (r):

20 CAπγ <≤ (B18)

We note that this is just one small subset of the infinite class of solutions to the Einstein equation, in particular the ones with cylindrical symmetry.

Note a): For A infinitesimal, we have a solution of the linearized Einstein equa-tions.

ENDNOTES

1) In a Euclidean-like structure of a frame, the Pythagorean Theorem is satisfied.

2) In support of Einstein’s covariance principle, it is often argued that the out-come of a calculation cannot depend on a choice of coordinates. However, a necessary implicit assumption in such an argument is that the physical inter-pretation of the coordinates does not change. The existence of the Euclidean-like structure in the frame of reference of a physical space (Lo 2003b) implies that the physical meaning of coordinates is necessarily gauge-dependent, and so are measurements. Therefore, the notion of gauge invariant “genuinely measurable quantities” is merely an illusion in mathematics. And the deflec-tion of light to second order is an example to Einstein’s covariance principle being invalid (Lo 2008a).

3) Few theorists in gravitation read beyond a few specialized journals to under-stand non-linear equations. Many are still unaware of the non-existence of dynamic solutions (Lo 1995, 2000b). Moreover, many insist on misinterpret-ing the formula E = mc2 to be unconditionally valid (Lo 1997, 2007d; Lo & Wong 2007), and ignore the challenge of the Royal Society (Bondi et al. 1959), who are earlier than Lo (1999a) and Zhou (1983) pointing out that there are inconsistencies in Einstein’s general relativity. Those theorists, including C. M. Will and also Eric J. Weinberg, editor of the Physical Review D, are out-dated,



167

about half a century. The root of the problems is essentially that they failed to understand Einstein’s equivalence principle (Lo 2008a).

4) Here, Weber & Wheeler (1957) have mistaken Einstein’s “Covariance princi-ple” as Einstein’s equivalence principle.

5) A common problem among some applied mathematicians is that they often take a conditionally valid mathematical expression as physically absolute and thus out of contact with the physical reality. This is a form of confusion on mathematics and physics. If Professor ‘t Hooft had paid more attention to the physics of plane-waves, he could have saved himself from making such an error. Moreover, if he studies more pure mathematics, he will be able to see that there is no dynamic solution and that Pauli’s version of equivalence prin-ciple is due to his errors in functional analysis, as illustrated by the example given by Einstein (Einstein et al. 1923; p. 144).

6) It was believed (Einstein et al. 1938; Feynman 1996) that the two-body prob-lem could be solved in Einstein’s equation. However, as suspected by Gull-strand (1921) and conjectured by Hogarth (1953), the opposite is correct (Lo 1995, 2000b).

7) Zhou (1983) pointed out, “The concept that coordinates don’t matter in the interpretation of Einstein’s theory … necessarily leads to mathematical results which can hardly have a physical interpretation and are therefore a mystifica-tion of the theory.” However, from Zhou’s experiment of local light speeds (Zhou 1987), it is clear that the term “coordinates” in the above statement should be understood as “meanings of coordinates” or “gauges”. This confu-sion could be avoided if such physical meanings have been clarified in terms of measurements (Lo 2003b).

8) Zhou (1987) proposed to measure the local light speeds. However, theorists such as Ohanian & Ruffini (1994), who do not understand Einstein’s equiva-lence principle, claimed that the measured light speeds are invariably c. The fact is that very small differences between light speeds of perpendicular di-rections have already been detected by the Michelson-Morley (1887) experi-ment (Miller 1933).

9) These two Princeton Professors failed to justify their assumed initial condi-tions for a dynamic case. It turns out that their assumption is not justifiable (Lo 1999a, 2000a).

10) However, as Burton Richter (2006) put it, “I have a very hard time accepting the fact that some of our distinguished theorists do not understand the differ-ence between observation and explanation, but it seems to be so.” It was very difficult to accept that some of our distinguished theoreticians cannot really tell the difference between mathematics and physics, but it is found to be so.


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168

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1. Bondi,H., Pirani,F. A. E. and Robinson,I.1959.Proc. R. Soc. London A .51:519.

2. Cheng,Tai-Pei .005, Relativity, Gravitation, and Cosmology – a basic intro-duction (Oxford University Press).

3. Christodoulou,D. & Klainerman,S. 1993. The Global Nonlinear Stability of the Minkowski Space (Princeton University Press). Princeton Mathematical series no. 41.

4. Eddington,A. S. 1975, The Mathematical Theory of Relativity (Chelsea, New York).

5. Einstein A. 1954, The Meaning of Relativity (Princeton Univ. Press).

6. Einstein,A., Infeld L., & Hoffmann,B. 1938. Annals of Math. 39 (1): 65-100.

7. Einstein,A., Lorentz,H. A., Minkowski,H.& Weyl,H. 1923, The Principle of Relativity (Dover, New York).

8. Einstein,A. & Rosen,N. 1937. J. Franklin Inst. 223: 43 (1937).

9. Feynman,R. P. 1996. The Feynman Lectures on Gravitation (Addison-Wesley, New York).

10. Fock,V. A.1964, The Theory of Space Time and Gravitation, translated by N. Kemmer (Pergamon Press).

11. G. ‘t Hooft 2007, claimed that his cylindrical symmetry solution is a wave packet (private communication, August and September 2007).

12. Gullstrand,A. 1921. Ark. Mat. Astr. Fys. 16, No. 8.

13. Herrera,L., Santos,N. O. & Skea,J. E. F.2003.Gen. Rel. Grav. 35, No.11, 2057 (2003).

14. Hogarth,J. E. 1953. Ph. D. Thesis, Dept. of Math., Royal Holloway College, University of London, p. 6.

15. Kennefick,D. 2004. “Einstein versus the Physical Review,” Phys. Today (Sep-tember).

16. Kramer,D., Stephani,H., Herlt,E., & MacCallum,M.1980, Exact Solutions of Einstein's Field Equations, ed. E. Schmutzer (Cambridge Univ. Press, Cam-bridge).

17. Landau,L. D. & Lifshitz,E. M. 1962. Classical Theory of Fields (Addison-Wesley, Reading Mass.).

18. Le,Bas Louise, Publishing Editor, the Royal Society, A Board Member’s Comments (July 24, 2007).



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19. Liu,H. Y. & Zhou,P.-Y. 1985, Scientia Sincia (Series A) 1985, XXVIII (6) 628-

637.

20. Lo,C. Y. 1995, Astrophys. J. 455: 421-428 (Dec. 20).

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29. Lo,C. Y. 2005, Phys. Essays, 18 (4): 539 (December).

30. Lo,C. Y. 2006a, Astrophys. Space Sci., 306: 205-215.

31. Lo,C. Y. 2006b, Special Relativity, Misinterpretation of E = Mc2, and Ein-stein’s Theory of General Relativity, in Proc. IX International Scientific Con-ference on ‘Space, Time, Gravitation,’ Saint-Petersburg, August 7-11, 2006b.

32. Lo,C. Y. 2007a, Bulletin of Pure and Applied Sciences, 26D (1): 29 - 42.

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34. Lo,C. Y. 2007c, Phys. Essays, 20 (3), (Sept.).

35. Lo,C. Y. 2007d, Chin. Phys. (Beijing), 16 (3): 635-639 (March).

36. Lo,C. Y. 2008a, Bulletin of Pure and Applied Sciences, 27D (1): 1-15.

37. Lo,C. Y. 2008b, The Mass-Charge Repulsive Force and Space-Probes Pioneer Anomaly, in preparation.

38. Lo,C. Y., Chan D. P., & Hui R. C. Y. 2002, Phys. Essays 15 (1): 77-86 (March).

39. Lo,C. Y. & Wong C. 2006. Bulletin of Pure and Applied Sciences, 25D (2): 109-117.

40. Michelson,A. A. & Morley E. W. 1887. Am. J. Sci. 34 : 333-345.

41. Miller,D. C. 1933.Reviews of Modern Physics 5: 203-241 (July).

42. Misner,C. W., Thorne K. S., & Wheeler J. A. 1973. Gravitation (Freeman, New York).

43. Norton,J. 1989. “What was Einstein’s Principle of Equivalence?” in Einstein’s Studies Vol. 1: Einstein and the History of General Relativity, eds. D. How-ard & J. Stachel (Birkhäuser).


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44. Ohanian, H. C. and Ruffini,R. 1994.Gravitation and Spacetime (Norton, New York).

45. Pauli,W. 1958, Theory of Relativity (Pergamon Press, London).

46. Penrose,R.1964. Rev. Mod. Phys. 37 (1): 215-220.

47. Richter,B. 2006. “Theory in particle physics,” Physics Today, October 2006:8.

48. Straumann,N. 1984, General Relativity and Relativistic Astrophysics (Springer, New York).

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52. Weinberg,S. 1972, Gravitation and Cosmology: (John Wiley Inc., New York), p.3.

53. Whitehead,A. N. 1922, The Principle of Relativity (Cambridge Univ. Press).

54. Will,C. M. 1981, Theory and Experiment in Gravitational Physics (Camb. Univ.).

55. Will,C. M. 1986, “Was Einstein Right?” (Basic Books, New York), p. 20.

56. Zhou,Pei-Yuan, 1983 “On Coordinates and Coordinate Transformation in Einstein’s Theory of Gravitation” in Proc. of the Third Marcel Grossmann Meetings on Gen. Relativ. ed. Hu Ning, Sci. Press & North Holland. (1983), 1-20.

57. Zhou(Chou), P. Y. 1987. “Further Experiments to Test Einstein’s Theory of Gravitation” in Proc. of the International Symposium on Experimental Gravitational Physics, Guang Zhou, China August.



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Dr Elisabeth A.L. Mol Editorial Director Foundations of Physics Springer Van Godewijckstraat 30 3311 GX Dordrecht Netherlands Phone:+31786576136 Fax +31 78 657 6377 E-mail:[email protected] Dear Dr. Mol: I am writing to you because I want to make sure that Professor G. ‘t Hooft would surely receive my paper, “THE PRINCIPLE OF CAUSALITY AND THE CYLINDRI-CALLY SYMMETRIC METRIC OF EINSTEIN AND ROSEN”, which will be published in Bulletin of Pure and Applied Sciences.Vol.27D (No.2) 2008:149 -170. This paper is my response to his detailed comments (which is faithfully included in the paper as Professor ‘t Hooft requests) on my earlier paper about the cylindrical symmetric metric of Einstein and Rosen. Since this paper is not in print yet, he is invited to give fur-ther clarification of his viewpoints if he feels desirable to give such additions.

Moreover, Professor ‘t Hooft has claimed in the INTERNET that I do not respond to comments of disagreement. This is simply not true and probably Professor ‘t Hooft was misinformed. Moreover, I want to let him know personally that this is a groundless misin-formation. About a month ago I have sent a copy of this paper to him to his email address. However, he has not acknowledged its reception yet. Moreover, it is also known that his email address does filter out unimportant emails. This is why I want to be sure he re-ceives this paper by going through you. Thank you.

The paper is related to an important issue in general relativity that Professor ‘t Hooft and I disagree. It is on whether Einstein's field equation has a solution for gravitational waves. Professor ‘t Hooft believes that such wave solutions exist whereas I have proven that no such wave solutions exist.

Initially, I also believed that such wave solutions exist since the linearized equation of Einstein has such wave solutions. However, in 1993 I have found that the linearization of Einstein equation, though valid for the static case, is not valid for the dynamic cases, from which the gravitational waves would be generated. Then, I start to investigate whether wave solutions exist for the non-linear Einstein equation, although I believe the physical existence of gravitational waves.

At first I find that there is no plane-wave solution and then I find that as suspected by Gullstrand in the report to the Nobel Committee [1], there is no physical solution for a two-body problem. This paper was published in Astrophysical Journal in 1995 [2]. In re-sponse to related questions, a supporting paper was published in Physics Essays in 2000 [3]. More recently, a conclusive paper on plane- waves is published in Astrophysical and Space Science [4].


C. Y. Lo

172

Apparently, Professor ‘t Hooft did not find any errors in the above papers. This is under-standable since Professor ‘t Hooft is essentially an excellent applied mathematician. In these papers, there are areas that he is not familiar with. Nevertheless, he believed that instead he could establish a counter example to support his viewpoint and it was sent to me. This approach of over simplification would inevitably lead to errors. (Recently a good example is that theorists of the black holes stand out against the experiment on high energy physics.) However, I appreciate his interest on this problem, but I respectfully dis-agree.

In my new paper, I show that THE CYLINDRICALLY SYMMETRIC METRIC OF EINSTEIN AND ROSEN violates both Einstein’s equivalence principle and the principle of causality. Therefore, this determined that the example of ‘t Hooft cannot be valid in physics. Moreover, his detailed calculations indicate that he does not understand adequately the physical nature of the plane-wave as an idealization. This amazes us since errors of such a nature normally should not occur to him.

For instance, when I was in an August Conference in London, people just could not be-lieve that Professor ‘t Hooft, a Nobel Laureate, could make such a mistake on the nature of plane-waves. Also, one of my friends Dr. C. C. Lo has to admit that Professor ‘t Hooft is in error after he failed to defense the position of Professor ‘t Hooft. Dr. C. C. Lo is a great admirer of Professor ‘t Hooft. He may recall that Dr. C. C. Lo has employed an art-ist to plaint a portrait of Professor ‘t Hooft and sent it to him.

Professor ‘t Hooft is now editor-in-chief of your journal. Such an important position de-mands almost perfection of a theorist, because his error would be decisive to your journal. I sincerely hope that Professor ‘t Hooft would improve himself such that he could serve your journal better.

Thank you for your kind attention. I am looking forward to hearing from you.

Sincerely yours, C. Y. Lo

References

1. Gullstrand,A. 1921. Ark. Mat. Astr. Fys. 16, No. 8.

2. Lo,C. Y. 1995, Astrophys. J. 455: 421-428 (Dec. 20).

3. Lo,C. Y. 2000b, Phys. Essays, 13 (4): 527-539 (December).

4. Lo,C. Y. 2006a, Astrophys. Space Sci., 306: 205-215.


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