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The work of Christodoulou on general relativity is based on obscure errors in mathematics, and gives misleading consequences. The problem of Einstein's equation was discovered by honorable Nobel Laureate Gullstrand, a member of the 1921 Nobel Committee. In 1955, Gullstrand is proven correct. Initially the error of Christodoulou is due to a failure in distinguishing the difference between mathematics and physics. Here the nature of his errors in mathematics and related literature are pointed out so that scientists can make better objective judgments. Also, examples that illustrate errors with mathematics at the undergraduate level, are useful since a similar mistake has already been made by the 1993 Committee for the Nobel Prize in Physics. It is found that many made judgments based on popular opinions, instead of the scientific evidence as Galileo advocates. In fact, blind acceptance of invalid claims has reached the level of Fields Medalists and Nobel Laureates.

In spite of Gullstrand's warning, before 1993 theorists including Nobel Laureates and Field Medalists, failed to see that linearization of the Einstein equation to obtain an approximate dynamic solution is not valid in mathematics. This error is manifested, with misinterpretation of Einstein's equivalence principle, in the 1993 press release of the Nobel Committee. In 1995 it has been shown that the Einstein equation cannot have a bounded dynamic solution for a two-body problem or gravitational wave solutions; and the Hulse-Taylor binary pulsars experiments actually support the modified Einstein equation. Nevertheless, due to biased misconceptions and earlier errors, many were unable to understand these mathematical results. Consequently, they failed to see the necessity of unification between electromagnetism and gravitation. To remedy this situation, two illustrative examples are shown. 1) For the claim of Misner, Thorne & Wheeler, having a bounded wave solution, their errors are illustrated explicitly with mathematics at the undergraduate level. 2) The unbounded ?weak plane-wave? of Bondi, Pirani, & Robinson violates the principle of causality.

It is reported that something has gradually dragged two of America's oldest space probes-Pioneer 10 and Pioneer 11- a quarter-million miles off course. This is called the pioneer anomaly. Astrophysicists have struggled for 15 years in vain to identify the infinitesimal force at play. It is shown that the anomaly should be due to a newly discovered force from the sun. Such a force is due to a charge - mass interaction, derived from general relativity. For a charge q and another particle of mass m, with a distance r between them, the static repulsive force is q²m/r³. Furthermore, such a force is not subject to electromagnetic screening, and this has been verified experimentally. Because of the r^-3 - dependence, the repulsive force becomes increasingly negligible as the distance r increases. This effect can be observed as the pioneer orbital anomaly. It is conjectured also that the anomaly of a planetary probe would be due to charge-mass interaction that includes the current-mass attractive force. Thus, all of the anomalies are related to the mass-charge interaction and thus are natural consequences of extending general relativity. Concurrently, it is pointed out that Einstein's equivalence principle is also crucial in understanding unification.  04.20.-q, 04.20.Cv

It has been found that it is impossible to have a dynamic solution for the Einstein equation and the nonexistence of gravitational wave solutions is due to a violation of the principle of causality. However, ?t Hooft claimed that his ?cylindrical symmetric wave? would challenge this conclusion. It is shown that the metric form of the Einstein-Rosen type violates the principle of causality. Thus, the ?wave? of ?t Hooft is invalid in physics because he fails to tell the difference between mathematics and physics. Moreover, there are two errors in his construction, namely: 1) the plane wave has been extended beyond its physical validity as an idealization, and 2) the integration over the angle is not a wave packet and cannot be justified with any physical process.

Einstein gives three predictions to support general relativity. However, the Einstein equation, which was first derived by Hilbert, has not been accurately confirmed beyond what the Maxwell-Newton Approximation can do. The gravitational redshifts, the bending light, and radar echo delay can be obtained from the Maxwell-Newton Approximation derived from Einstein's equivalence principle. As Gullstrand suspected, it is proven that there is no dynamic solution and thus the perihelion of Mercury cannot be derived from the Einstein equation. Moreover, the Hulse-Taylor experiment of binary pulsars actually supports a modified Einstein equation. So far, the only exception is the derivation of the metric for a charged particle because the electromagnetic energy-stress tensor is involved. It has been shown that this metric implies a repulsive force mq²/r³ between a charge q and a mass m, separated with a distance r. Thus, the experimental confirmation of this neutral force provides the only case to verify the static Einstein equation. Although it has been shown experimentally that a metal ball becomes lighter after charged with electrons, more detailed data are needed to confirm the repulsive force and distinguish its formula from claims of other theories.

The Nobel Prizes were established in 1895 by the Swedish chemist Alfred Bernhard Nobel for those who confer the "greatest benefit on mankind", and specifically in physics, chemistry, peace, physiology or medicine, and literature. In 1968 the Nobel Memorial Prize in Economic Sciences was established. Since the first Nobel Prize was awarded in 1901, the proceedings, nominations, awards, and exclusions have generated criticism and controversy. Here, the controversies and errors related to the Nobel Physics Prize are discussed. It is concluded that, to implement the will of Nobel better, it is necessary to do some subsequently following up works after the prizes are awarded. Moreover, it is suggested: 1)A s
the will of Nobel demands, the Nobel Committee should formally rectify their past errors in sciences. 2) Timely update the status of achievements awarded Nobel Prizes in Physics, Chemistry, and Physiology or Medicine when clarifications are necessary. This would further enhance the service to the will of Nobel. 3) To strengthen the implementation of Nobel's will, a Nobel Prize for Mathematics should be established.

Recently, calculation of the deflection angle to the second order also shows gauge invariance in mathematics. Nevertheless, careful analysis shows that this calculation actually implies that the theory is intrinsically not gauge invariant since, for each gauge, the shortest distance r₀ from the sun center is different from that for another gauge. Some argued that r₀ is just a label, but not a physical quantity. This is directly in conflict with Einstein's calculation that the deflection angle is 4KM/r₀, where M is the total mass of the sun and K = G/c² = 7.425x10^-29 cm/g. Thus, r₀ is not just a label. Hence, Einstein's covariance principle is intrinsically not valid in physics. Thus, logical maturity is currently a major problem in general relativity.

Einstein's accurate predictions created a faith that makes a critical analysis of general relativity over due. Since his field equation has no dynamic solutions, the observational confirmations have been exaggerated. Einstein's covariance principle has been proven to be invalid. This error comes from Einstein's theory of measurement that adapts the mathematical notion of distance in Riemanian geometry as if valid in physics, and his supporting arguments are actually based on invalid applications of special relativity. However, such a theory of measurement was not used in Einstein's predictions. Nevertheless, Einstein's equivalence principle plays a crucial role in rectifying the shortcomings of his theory, and the Maxwell-Newton Approximation is proven as independently valid for massive sources. Then, general relativity leads to the discovery of the charge-mass interaction that would explain the space-probe pioneer anomaly discovered by NASA. Thus, unification of gravitation and electromagnetism is proven necessary. Moreover, since the photons must include gravitational energy, particle physics would not be clearly understood without gravity.

Due to Einstein?s ?covariance principle?, Einstein?s requirement for weak gravity has not been universally accepted although many believe in both his requirement and principle. For example, Bondi, Pirani, & Robinson rejected Einstein?s requirement since they considered their unbounded metric as a weak wave. However, validity of this ?wave? actually has not been established since the related metric for no gravity clearly violates the principle of causality. Some theorists rejected Einstein?s requirement for being coordinate-dependent because they do not understand that it is based on the principle of causality. However, based on Einstein?s equivalence principle and related Einstein-Minkowski condition, such a coordinate-dependence is solved. Validity of this principle and condition are illustrated with the metric of Einstein?s rotating disk. Concurrently, Einstein?s requirement is proven valid. Moreover, invalidity of Einstein?s ?covariance principle? is illustrated with a counter example.

It is shown that Einstein?s proof for E = mc² is actually incomplete and therefore is not yet valid. A crucial step is his implicit assumption of treating the light as a bundle of massless particles. However, the energy-stress tensor of massless particles is incompatible with an electromagnetic energy-stress tensor. Thus, it is necessary to show that the total energy of a light ray includes also non-electromagnetic energy. It turns out, the existence of intrinsic difference between the photonic and the electromagnetic energy tensors is independent of the coupling of gravity. Nevertheless, their difference is the energy-stress tensor of the gravitational wave component that is accompanying the electromagnetic wave component. Concurrently, it is concluded that Einstein?s formula E = mc² necessarily implies that the photons include nonelectromagnetic energy and that the Einstein equation of 1915 must be rectified.

The theoretical foundation of LIGO?s design is based on the equation of motion derived by Thorne. His formula, motivated by Einstein?s theory of measurement, shows that the gravitational wave-induced displacement of a mass with respect to an object is proportional to the distance from the object. On the other hand, based on the observed bending of light and Einstein?s equivalence principle, it is concluded that such induced displacement has nothing to do with the distance from another object. It is shown that the derivation of Thorne?s formula has invalid assumptions that make it inapplicable to LIGO. This is a good counter example for those who claimed that Einstein?s equivalence principle is not important or even irrelevant.

Currently, Hubble?s law is often considered as the observational evidence of an expanding universe. It is shown that Hubble?s Law need not be related to the notion of Doppler redshifts of the light from receding Galaxies. In the derivation of the receding velocity, an implicit assumption, which implies no expansion, must be used. Moreover, the notion of receding velocity is incompatible with the local light speeds used in deriving the light bending. The notion of an expanding universe is based on an unverified assumption that a local distance in a physical space is similar to that of a mathematical Riemannian space embedded in a higher dimensional flat space, and thus the physical meaning of coordinates would necessarily depend on the metric. However, this assumption has been proven as theoretically invalid. In fact, a physical space necessarily has a frame of reference, which has a Euclidean-like structure that is independent of the yet to be determined physical metric and thus cannot be such an embedded space. In conclusion, the notion of an expanding universe could be just a mathematical illusion.

It is pointed out that Special Relativity together with the principle of causality implies that the gravity of an electromagnetic wave is an accompanying gravitational wave propagating with the same speed. Since a gravitational wave carries energymomentum, this accompanying wave would make the energy-stress tensor of the light to be different from the electromagnetic energy-stress tensor, and thus can produce a geodesic equation for the photons. Moreover, it is found that the appropriate Einstein equation must additionally have the photonic energy-stress tensor with the antigravity coupling in the source term. This would correct that, in disagreement with the calculations for the bending of light, existing solutions of gravity for an electromagnetic wave, is unbounded. This rectification is confirmed by calculating the gravity of electromagnetic plane-waves. The gravity of an electromagnetic wave is indeed an accompanying gravitational wave. Moreover, these calculations show the first time that Special Relativity and General Relativity are compatible because the physical meaning of coordinates has been clarified. The success of this rectification makes General Relativity standing out further among theories of gravity.