Rationality Probles of the Principles of Equivalence and General Relativity Mei Xiaochun (Departent of Physics, Fuzhou University, E-ail: xc1@163.co Tel:86-591-8761414) (N.7-B, South Building, Zhongfu West Lake Garden, Fubin Road, Fuzhou, 355, China) bstract It is pointed out that at present we only prove that inertial static ass and gravitational static ass are equivalent. We have not proved that inertial oving ass and gravitational oving ass are also equivalent. It is proved by the dynaic effect of special relativity that inertial oving ass and gravitational oving ass are not equivalent. Besides, it can be proved that the principle of general relativity is untenable. It is only an apparent felling of ankind actually. Therefore, there exists serious defect in the foundation of the Einstein s theory of gravitation. We need to renovate our basic idea of space-tie and gravity. PCS Nubers: 4..-q 3.3.+p 4..Cv 4.9.+e 4..Dw Key Words: Equivalent Principle, Special Relativity, General Relativity, Gravitation, Inertial Force Inertial Moving Mass Gravitational Moving Mass Equivalent principle is not consistent with special relativity We point out at first that the equivalent principle is not consistent with special relativity. Einstein used the reference frae of rotational desk to show the principle of equivalence and concluded that the space-tie of gravitational field was curved. s shown I Fig..1, suppose that the desk with radiu r and perieter π r is at rest in beginning. Then let the desk turn around its center in a unifor angle speed ω. ccording to the contraction of space-tie contract in special relativity, observers on the resting reference frae of ground would find that the perieter of desk becoes π r 1 ω r / c K owing to desk s rotation, but the radiu of desk would not contract for there is no otion in the direction of radiu. Therefore, the radio of perieter and radiu would be less than π and the observers on the ground would affir that the space of rotating desk is non-euclidean space. K 1 Fig..1 Fig.. Fig..3 1
The proble now is what the observer resting on the rotating desk thinks about. Einstein and alost textbook of general relativity took a istake here. It were though that because the speed of tangent direction caused length contraction, when these observers used their rulers to easure the perieter of desk, the perieter becoes π r / 1 ω r / c, longer than that of resting desk. But when they use their rulers to easure the radiu of desk, the length is unchanged. So the ratio of perieter and radiu is big than π. However, this result is wrong, for it is based on the preises that desk is at rest and observer and his ruler ove. But in this case desk rotates actually so that the perieter of desk and the rulers of observers contract synchronously. So the observers who are at rest on desk can not find the change of desk s perieter actually by using their rulers. Secondly, let s show that the principle of equivalence contradicts with the concept of space-tie relativity. There are two probles here. The first is that according to the coon understanding of special relativity, space-tie s contraction is caused by relative speed, which has nothing to do with acceleration ( ) 1. Or speaking strictly, the effect of acceleration is too sall coparing with the effect of speed so that it can be neglected. So according to space-tie relativity, for observers resting on rotating desk, is at rest but ground reference frae oves around K shown in Fig... In this way, the observers resting on K 1 desk would think that the rulers on their hands are unchanged so that they can not conclude that space-tie of desk is curved. Instead, they would think that space-tie of ground reference frae should be curved for is oving around K. Of course, this result can not be accepted. Therefore, if the concept K 1 of space-tie relativity holds, we can not conclude that the space-tie of gravitational field is curved. Conversely, if the space-tie of gravitational field is curved and the principle of equivalence holds, the principle of space-tie relativity would be violated. The second proble is that in light of general recognition of special relativity, space-tie contract is a purely relative effect, having nothing to do with acceleration. But according to the equivalent principle, space-tie contraction should be relative to force and interaction, not a purely relative effect. It is obvious that the principle of equivalence contradicts with the principle of space-tie relativity. Because the principle of equivalence is regarded as the foundation of the Einstein s theory of gravitation it can be said that special relativity is not consistent with the Einstein s theory of gravity. Only way for us to get rid of this paradox is to give up the relativity of space-tie, and adit that space-tie contraction is a kind of effect caused by accelerating processes. The rational result should be that the perieter of rotating desk contracts and the radio of perieter and radiu is less than π. No atter who are on rotating desk or on ground, observer s viewpoints are the sae.. Inertial oving ass and gravitational oving ass are nonequivalent Let s discuss the proble of equivalent principle itself. The principle can be divided two parts, one is weak equivalent principle and another is strong equivalent principle. The weak equivalent principle indicates that gravitational ass is equivalent with inertial ass, or gravity is equivalent to inertial force locally. We fist discuss the equivalence of gravitational ass and inertial ass. Then discuss the equivalence of gravity and inertial force, as well as the probles of space-tie contraction in gravitational field and non-inertial reference frae and so do. t present, the experients to prove the equivalence between inertial ass and gravitational ass is the so-called E otvos && & type of experients. It can be pointed out that this type of experients can not expose the equivalence of inertial oving ass and gravitational oving ass actually. ccording to the second law of Newtonian, the force acting on an object is F = a. Here is inertial static ass and i i K 1 K
a is acceleration. In a unifor gravitational field, the forula of force can also be written as Here F g = g. is gravitational static ass and g = constant is the strength of gravitational field. So we have g g a = g.1 If gravitational resting ass is equivalent with inertial resting ass, the radio i = g i should be a constant for any aterial. In this way, the acceleration of an object falling in this unifor gravitational field would also be a constant. This result has an obvious defect, i.e., the object s speed would surpass light s speed in vacuu at last when the object falls in the gravitational field. In order avoid this proble, the dynaic relation of special relativity should be considered. We have i = i 1 V / c F d dt V i i = =. a ( 1 V / ) 3/ 1 V / c c Because the effect of speed on gravitational force is still unknown at present, we suppose ( V ) = f.3 g g The function f ( V ) is waited to be deterined. Suppose we still have F = g, Eq.(.1) becoes g 3/ ( ) ( / a 3 / V = g gf ( V ) 1.4 c i s long as f V 1 V c ), the acceleration is not a constant again. When V c we ay have a, so that the falling speed would not surpass light s speed. It is obvious that when an object oves in a gravitational field, the effect of special relativity should be considered. In the E otvos && & type of ( ) experients, the oent acting on the hang bar is B i i T = l at g.5 B g g Here l is the length of bar, a is the centrifugal acceleration on the level direction caused by the i B i earth s rotation, and are the inertial asses, and are the gravitational asses of two different aterials and B individually. If the dynaic effect of special relativity is considered, by using forulas (.) and (.3) Eq.(.5) can be written as T T l a 1 V It is obvious that T has nothing to do with ( V ) g i i g g B g B T g = B / c.6 f. s long as the radio / i g does not change with different aterial, no atter what is the for of function f ( V ), we always have T =. It eans that the current experients of the E otvos && & types only verify the equivalence of gravitational static ass and inertial static ass without verifying the equivalence of gravitational oving ass and inertial oving ass. The relation between the will be deduced in later paper. On the other hand, the weak principle of equivalence can also be described as that gravitation is locally equivalent with inertial force. Besides rotating desk, Einstein used the ideal experient of closed chaber to show the equivalence. In the experients, observers in a closed chaber could not decide 3
whether they were at rest in a unifor gravitational field, or were being accelerated with a unifor acceleration. This result is correct in the Newtonian theory, but can be proved to be incorrect after the dynaic effect of relativity is considered. s shown in Fig..4, suppose that the chaber K is accelerated relative to resting reference frae K. There is an object with static ass is hanged on the ceiling of chaber through an elastic rope. When K is accelerated in a unifor acceleration, owing to the dynaic effect of special relativity as shown in Eq.(.), the inertial ass of object would becoe bigger and bigger. So the rope would be pulled longer and longer besides the Lorentz contraction. When the speed of chaber is great enough, the inertial force of object would surpass the bearing capacity so that the rope would be pulled apart. Because the breaking of rope is an absolute event, observers who are inside and outside chaber would observe it. However, if observers were at rest in a static gravitational field, the rope would not be pulled longer and longer and break at last. The observers inside chaber would affir that they are accelerated, instead of resting at a static gravitational field. It is obvious that the equivalence of gravitation and inertial force is only tenable in Newtonian echanics. fter the dynaic effect of special relativity is considered, gravitation and inertial force are not equivalent locally. Fig..4 Nonequivalence of gravitational oving ass and inertial oving ass However, if the concrete for of function f ( V ) is deterined, under the proise that Eq.(3.4) is satisfied, we can still say that the gravitation and inertial force are still equivalent at a certain extent. The force acted on an object located on a rotating desk can be equal to a gravitation which does not change with tie. The force acted on an object located in an accelerated and closed chaber can be equal to gravitation which changes with tie. In fact, as we show later that gravitation is only a kind of special non-inertial force actually. The equivalent probles of space-tie contractions in gravitational fields and non-inertial reference fraes are discussed below. Einstein used the ideal experient of rotating desk to shown that the space-tie of gravitational field was curved based on equivalent principle. We now discuss this proble. Because the clocks on rotating desk has a oving speed, its tie would becoe slow with t r ω = t.7 c 1 1 Here 1 t is the tie of clocks on static ground. ccording to the principle of equivalence, as shown in Fig..., the reference frae of uniforly rotating desk is equivalent with a static gravitational field which does not change with tie. In the figure, F is the centripetal force acting on clock, F is inertial force. The equilibriu of two forces keeps the clock resting on the desk. ccording to the equivalent principle, as shown in Fig..3, F should be regarded as a certain force just as frictional force and so do. This force balances gravitation G so that clock can be at rest in gravitational field without falling down. So as long as the equivalent principle is tenable, the tie 4
delay of clock which is at rest in gravitational field is expressed by Eq.(.7). In the 6 s last century, the experients of atoic cesiu clocks circling the earth had proved that clocks would becoe slow in gravitational field. They can be considered as the direct proofs that the existence of gravitational fields would cause tie s delay. But there still exist a fundaental proble as shown below. In atoic cesiu clock, tie is deterined by atoic vibrations. The fact that atoic clock becoes slow in gravitational field indicates that the frequency of atoic vibration becoes slow, or the frequency of photon radiated by ato becoes slow and wave length becoes long or red shift. ccording to the theory of curved space-tie, aterial s existence causes space-tie curved. In general relativity, there is no the concept of force. Particles are not acted by force when they ove along the geodesic line in curved space-tie. Therefore, there is no the concept of potential energy for a photon oving in gravitational field. The total energy of a photon is equal to its dynaic energy. On the other hand, if the forula E = hν is always tenable at any point of gravitational field and the frequency ν is not a constant, E would change at different place in gravitational field. In this way, the law of energy conservation of photon would not be violated. However, the violation of energy conservation is unacceptable, though physicists see to evade this proble at present. It will be shown in later discussion that we can establish gravitational theory in flat space-tie. In the theory, photon would be acted by gravitation so the concept of potential energy is eaningful. Based on the theory, we can still reach the sae result that gravitational field would cause tie delay without considering the principle of equivalence as a foundation for gravitational theory. Now let s discuss the proble of space bending based on the equivalent principle. ccording to the current theory, when the angle speed of rotating desk is ω, the space etric of desk is ( 3) r d + 1 r ω / c σ = dr + dϕ dz.8 It eans that length contraction takes place along the tangent direction of desk. Because there is no speed along the radiu direction, there exists no length contraction in this direction. Otherwise, according to the sphere syetrical solution of the Einstein s equation of gravity, the space part of Schwarzschild etric can be written as 1 α = 1 dr + r sin θ dϕ r dθ d σ + r.9 Because the slice of sphere is just the desk, so the situation of space bending should be the sae. In fact, fore thin desk, we can let dz = in Eq.(.8). For the slice of sphere, we can let θ = π / and d θ = in Eq. (.9). In this case, Eqs.(.8) and (.9) should describe the sae space etric. How, as we see that Eqs.(.8) and (.9) are not consistent in this case. In the Schwarzschild etric, length contraction takes place in the direction of radiu, which is just the direction of gravitation. But in the etric of rotating desk Eq.(.8), length contraction takes place in the tangent direction of desk, instead of the direction of inertial force. It indicates that space bending caused by non-inertial force ay be different fro that caused by gravitation, if gravitational fields would also cause space bending really. Unfortunately, this obvious difference has not caused person s attention up to now. It is obvious that even though the principle of equivalence holds so that gravity would cause space bending, the influences of non-inertial force and gravity on space ay be different. In fact, it is lack of direct proofs to prove that gravity would cause space bending. What we have now is only indirect proofs. We only prove it by calculating object s otions in the gravitational field of the sun. The result is equal to space s bending at a certain degree. But they are indirect 5
proofs and we need direct proofs. Even direct proofs prove that gravity would cause space bending in future, as shown in Eq.(.9), the results ay be different fro that caused by non-inertial force. In this proble, further theoretical research and experients are needed. In fact, in the theory of this paper, we can establish rational gravitational theory in flat space-tie without the concept of curved space. Besides, there are any other experients to show the ipossibility of weak equivalent principle. For exaple, when a closed chaber is at rest on a uniforly rotating desk, there exists the Coriolis force for a oving object, the inertial gyroscope would change its direction and charged objects would radiate photons. But in a static gravitational field, there are no such phenoena. Observers can distinguish both to be on rotating desk or at rest in a unifor gravitational field by theses phenoena. The proble of weak equivalence is very coplex. Soetie it is effective, soetie it is wrong. Soetie it contains soething specious. So it is iproper to take it as a fundaental principle of physics. 3. Ipossibility of the principles of strong equivalence and general relativity The strong equivalent principle can be described as that for any space-tie point in a gravitational field by taking the local inertial reference frae, the fors of nature laws are the sae with that when the Descartes reference frae is taken without gravitational field and acceleration. Or speaking directly, in a local reference frae which is falling freely in a gravitational field, gravitation would be eliinated. This sees to be a well-known experiental fact. Then, is the strong equivalent principle tenable? The following analysis shows that this is only an apparent feeling of ankind and is wrong actually. In order to explain it, we need to establish the concepts of integral force and non-integral force. So-called integral force indicates that the force acted on an object is distributed uniforly over all parts of the object, even over every atos and eleental particles of the object. Non-integral force indicates that the force is only acted on a part of the object. It is obvious that gravitation is integral force, but the force acted on an observer who is at rest in non-inertial reference frae is non-integral force. For exaple, when an observer stands on an accelerated train, the force acted on hi actually acts on his feet and the feet draw the body of observer oving forward. The other parts of body, not being acted directly by the force, are still in internal state, so that the body oves backward. In this way, so-called inertial force is caused. In an accelerating elevator, the force is only acted on observer s feet directly. Then the force is contributed to whole body through feet. For observers on rotating desk, friction force between feet and desk is need so that the observer can stand on desk. The friction force is also acted on feet directly. ll of these forces are non-integral forces. It is just owing to the action of this kind of non-inertial force, non-equilibriu or internal stress is caused in observer s body so that observer can feel the existence of accelerating otion. It should be pointed out that integral force is only a acro-concept. Such as friction force, the essence of non-integral force is electroagnetic force. Fro icro-angle, however, all interactions between icroparticles are integral. It is clear that the force acted on an object in unifor gravitational field is integral one. The inertial force caused by non-inertial otion is also integral force in general. The proble is now that when a non-integral force is distributed over whole body of observer who is at rest in a non-inertial reference frae what would happen? s shown in Fig. 3.1, the centripetal force acted on the feet of observer who is at rest on rotating desk is friction force actually. If this friction force is distributed over each part of observer s body, the observer would not fell this force s action. We can see in Fig.3. fro the angle of non-inertial reference frae, this uniforly distributive friction force would be uniforly counteracted by the uniforly distributive inertial force. In this case, the observer who is at rotating desk sees to be at rest 6
in a static inertial reference frae without any force s action, though he is actually accelerated. He ay also think that he is falling in a gravitational field, just as an astronaut travels in the orbit of cincturing earth. So it is only a subjective feeling that observer was not acted by force when he fell freely in a unifor gravitational field. The true is that the force acted on observer s body is integral force which is distributed over observer s body uniforly. In this way, no internal stress is produced in observer s body so that he can not fell the action of gravitational force and the existence of acceleration. Physiology tells us that felling is caused by non-unifority. No felling does not indicate no changing. If changing is unifor and slow, there would be no felling. If proper ethods are used, surpassing subjective felling, the observer would find hiself to be being accelerated when he falls freely in a unifor gravitational field. For exaple, a charged object would radiate photons when it falls freely in a gravitational field, but the object would not when it is at rest in inertial reference frae. So only by taking a charged object with hiself, the observer would judge whether he is falling freely in a unifor gravitational field or at rest in inertial reference frae. Only by this fact, the principle of strong equivalence has been proved untenable. More iportant is that coparing with static inertial reference frae, oving ruler s length would contract and oving clock s tie would becoe slow as well as oving object s ass would increase absolutely. The fors of nature laws are different in both the freely falling reference fraes of gravitational fields and in the Descartes inertial reference frae. Einstein denied the existence of absolutely resting reference frae by establishing special relativity, thought that all inertial reference fraes were equivalent for the description of nature phenoena. fter that, he introduced the principle of general relativity again to eliinate the superiority of inertial reference frae, thought that all reference fraes with arbitrary oving fors were equivalent for the description of nature phenoena. In atheatics, the principle of general relativity can be described as that the basic fors of otion equations should be covariant, or the line eleent of arbitrary reference fraes ust satisfy relation ds = constant. This deand is rational, for it actually represents the general for of the invariability principle of light s speed under the condition of non-inertial transforations. But it does not ean that the fors of forces acted on objects are the sae. s we known, in the general three diension space, the otion equations can be transfored into other fors by introducing arbitrary coordinate transforations without bring any practical physical result, for there are no any force or acceleration is introduced. But in the four diension space-tie, the situation is copletely different. In the four diension space-tie, if the transforation is linear just as the Lorenz transforation, velocity would be introduced. But if the transforations are arbitrary, accelerations or non-inertial forces would be introduced in otion equations. In this way, there still exists a superior reference frae actually, i.e., inertial reference frae, in which otion equation has siplest for without the existence of inertial force. In the Einstein s theory of gravity, non-inertial otions are considered to be equal to gravitational fields. In this situation, any probles are caused. t present, based on the principle of general relativity that any reference frae is equivalent to describe a physical syste, when the coordinate transforation is carried out for a solution of gravitational field s equation, it is thought that the solution with new for would still be effective to describe the original gravitational field. However, even according to the principle of equivalence, this is also ipossible. If non-inertial reference frae is equivalent locally with gravitational field, a special non-inertial reference frae can be only equal to a special gravitational field, with the corresponding relation of one by one. So when a non-inertial reference frae is transfored into another one, it eans that a gravitational field is transfored into another one with different nature. That is to say, because inertial forces are considered to be equal to gravities, the general transforation in the 7
space-tie of four diensions would change gravitational fields. For a certain gravitational field, suppose we have obtained its solution by solving the Einstein s equation. If the coordinate transforation is carried out, the for of field s equation and the solution are changed. The result eans that the original field has been transfored into new gravitational field, and new solution would not be original one. So a certain gravitational field can only correspond to a certain etric. rbitrary coordinate transforation in the space-tie of four diensions is not allowed, unless in new reference frae, the sae result can be obtained. However, this is ipossible in general. For exaple, we can not re-calculate the perihelion precession of Mercury as well as other probles in the Leaite and Kruskal coordinate systes and get the sae result as we do in the Schwarzschild coordinate syste. In fact, the reason that the energy-oentu tensors of gravitational field can not be defined well at present is actually owing to the existence of strong equivalent principle. ccording to the principle, we can always introduce sae local inertial reference fraes through coordinate transforations, in which gravitational fields would be eliinated and the energy-oentu tensors of gravitational fields also becoe zero. However, according to the law of energy-oentu conservation, a static syste s energy and oentu should be constants. How can they be canceled only by coordinate transforations? Therefore, it can be said clearly that no relativity and arbitrarily are allowed in the description of gravity. The description of gravitational theory deands absolution. The principle of general relativity is untenable. The deand that the fors of otion equations should be covariant in arbitrary reference fraes is only a basic restriction for the transforations of otion equations. It does not indicate that the descriptions of physical processes are the sae in arbitrary reference fraes. In fact, only in inertial reference fraes, the basic fors of otion equations can be the sae. nd as entioned in the paper bsoluteness of Velocity Produced by ccelerating Process and bsolute Space-tie Theory with Variable Scales, only in the absolute reference frae, the descriptions of physical processes are the siplest and ost syetrical. t last, we should point out that the free falling of reference fraes in gravitational field is only general accelerating otion. The force causing acceleration is only special one---- gravity. The result is that relative to resting reference frae, in accelerating reference frae, length would contract and tie would delay and ass would increase. ll of those are the noral effects of special relativity. Therefore, as long as we know the interaction for between gravitational field and aterial, we can describe object s otions in gravitational fields by the dynaic ethod of special relativity. The description is in flat space-tie. It is unnecessary to us to introduce the equation of gravitational fields in curved space-tie. This proble will be discussed in next paper. It is obvious that though the Einstein s theory of gravity has reached great success, there are any probles existing in its logical foundation and theoretical configuration. We should renovate our ideas about the essence of space-tie and gravity. References 1. Zhang Yuanzhong, Experient Foundations of Special Relativity, Science Publishing House, 87 (1997).. S.Weinberg, Gravitation and Cosology, Science Publishing House, 13 (1984). 3.Zhang yongli, Introduce to Relativity, Yunnan People Publishing House, 77 (198. 4. Sun Zhiing, Tensors in Physics, Beijing Noral University Publishing House, 18 (1985). 8