The gravitational phenomena without the curved spacetime

Transcription

1 The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime, whih merges the Minkowski spaetime and the mass density into the single idea. The new desription of the gravitational phenomena is kept in the spirit of Mah, in opposite to the Newtonian gravity and GR. All phenomena of gravity are epressed in terms of the relationship between a bodies and not between the body and surrounding the spaetime. keywords: general relativity; modified gravity; gravitational waves Introdution General Relativity (GR) is a theory, whih a sine about 00 years desribes the gravitational phenomena as a geometri properties of the four-dimensional spaetime. Although GR is widely aepted as a fundamental theory of gravitation for the many physiists still this is not a perfet theory. In GR the spaetime plays a very important role as the medium. The spaetime ontinuum is the mathematial model, that merges the three-dimensional spae and one dimension time into a single idea - the four-dimensional spaetime. Under influene outer gravitational field the four-dimensional spaetime is urved. Although the GR survived so far all tests observation, diret detetion of the gravitational waves still remains missing part of the puzzle. In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime, whih merges the Minkowski spaetime and the mass density into the single idea. The physial onsequenes of a suh replaement are a very interesting and far-reahing. The alternative desription of the gravitational phenomena The arena, where gravitational phenomena take plae is the ontinuous medium, immersed in the Minkowski four-dimensional spaetime, whih is the only the bakground. The medium is infinite olletion of material bodies filling the whole spaetime about a ertain mass density, whih has the apaity to propagate the gravitational interations. Let us assume that in the absene of the outer gravitational field, the medium beomes the medium. This medium, with the mass density, is homogeneous, isotropi, independent of the time and is defined as follows def = η = diag(,,, ) () where: is the mass density tensor, η is the Minkowski tensor, µ, = 0,,, 3. This mathematial model merges the Minkowski spaetime and the mass density into the single idea. This is way out beyond the well-known sheme.

2 Note that never reahes zero, although it may be very lose. The medium is never empty, in the ontrast to the vauum. For eample, the mass density tensor in the spherial oordinates has form ( r) = diag(,, r, r sin θ) omponents: 00 and rr has dimension [kg m -3 ], but omponents φφ and θθ [kg m - ]. Under influene outer gravitational field the medium with the mass density and deformed and beomes the effetive medium with the effetive mass density tensor. is stressed. The metri The metri in the effetive medium is defined as ds ( ) def = µ d d () where: is a symmetri, position dependent, the effetive mass density tensor. Tensor desribes the mathematial relationship between the effetive medium and the medium under g. influene gravitation and, in a some sense, is equivalent to the metri tensor ( ) The tensor desribes all the properties of the effetive medium and establishes metri, whih is desribed by the equation (). The gravitational field hange the gravitational field, the metri () beomes the Minkowski metri. Let us onsider the spaetime with metri µ ( g ) g d ds = d and also metri (). In absene of and metri of the effetive medium ().. Postulate All gravitational laws in urved spaetime and in the effetive medium are the same. This is an etension of the Priniple of Relativity, whih means, that the mathematial strutures desribing all gravitational phenomena in two different media are the same. We epet that they must lead to the same empirial data in the both different media (we assume also, that = ), therefore ds ( ) = ds ( g ) and finally we get the mathematial bridge between these the two different media in form

3 = g. (3) This mathematial relationship will be helpful in finding the field equations in the effetive medium..3 Building a bridge between different media Einstein s field equation takes the form 8πG ( g ) g R( g ) = T ( g ) R where physial quantities: R ( g ) the Rii tensor and ( g ) g, tensor ( ) 3 urved spaetime, both are the funtion of the metri tensor tensor, G is the gravitational onstant, is the speed of light. () R the Rii salar are desribing the T is the stress energy g The left side of the equation () desribe the Riemann geometry and has the physial dimension [m - ], while the right side of the equation () desribe the material soure, and without oeffiient, has the physial dimension [N m - 8πG ]. The oeffiient has the physial dimension [N - ] and joins two different worlds, the world of the Riemann geometry with the physial world of the energy, momentum and pressure. When the stress energy tensor disappears, i.e. T ( ) = 0 loses in itself, what an lead to the singularity. g, then the world of the Riemann geometry Aording to Postulate, when the urved spaetime we will replae with the effetive medium then, the general form of the field equation () will not hange σ ( ) σ( ) ~ T ( ) The new physial quantities: σ ( ) the stress tensor of the effetive medium and ( ) (a) σ the stress salar of the effetive medium are desribing the effetive medium during the stress. The stress salar σ σ. All physial quantities are now the of the effetive medium an epress as ( ) ( ) funtion of the effetive mass density tensor. Both sides of the equation (a) have now the same physial dimension [N m - ] and desribe the same physial world. The gravitational field interats with the medium, ausing that this medium is stress and beomes T desribes the soure of these stress. The the effetive medium. The stress energy tensor ( ) stress tensor of the effetive medium σ ( ) desribes the magnitude of these stress. Under the influene of these stress the effetive medium is strain. We suppose also, that σ ( ) ~ R ( g ) and σ( ) R( g ) ~. This mathematial onsiderations will help us to find the equation of the field.

4 . The field equation We searh for the field equation of the seond order in the field variables, whih, to be onsistent with the observations, and in the weak field and slow motion limit (Newtonian limit), must redue to the lassial Poisson equation for the lassial gravitation. The Lagrangian density L eff for the effetive medium should ontain the field variables and their first derivatives only. The field variables are desribing the effetive medium. In agreement with these requirements, we assume that where: det( ) L eff = 6 π σ = is the determinant of the effetive mass density tensor. We write the total ation over an arbitrary the effetive medium region Ω as ( L + L ) eff m S= d Ω where: L m is the Lagrangian density for the matter and fields. The term volume hange in the effetive medium, while d medium. After substituting, the total ation has form S= σ + Lm d 6π d desribes the 0 3 = d d d d desribes the volume in the After the mathematial alulations well-known from GR, aording to the equation (3), the searh of the field equation will get the orret form σ ( ) σ( ) = 8π T ( ) (5) where L ( ) m L T + m. The soures and fields in the equation (5) are the same and not differ from eah other (see also the equation (7)). Were obtained a new understanding of phenomena of gravity. The distribution of the mass density in the Universe T ( ) determines the stress of the effetive medium in the whole Universe. Please note, that T ( ) never reah zero. Paraphrasing John Arhibald Wheeler's words, we an say that: matter tells the effetive medium how to stress, the effetive medium under the stress tells matter how to move.

5 The field equation (5) has eatly the same form as the Einstein s field equation (), if we will use the following relations σ σ = g ( ) = R ( g ) ( ) = R( g ) G G (6) The oeffiient =. 0 N and is a probably the greatest fore in the Universe. It tells us about G the sale of the transition between these two different worlds..5 Correspondene with Newtonian theory of the gravitation In the Newtonian approimation we an deompose to following simple form where << = +, is a very small perturbation in the effetive mass density tensor, then the field equation (5) takes the form πg 8 00 (7) The distribution of the mass density (the soure) determines the distribution of the effetive mass density (the field) around the soure. 00 The equation (7) is a speial ase, for ( ) lassial gravity where V is the gravitational potential. 00 V πg, the well-known Poisson equation for the V (8).6 The equations of motion The Lagrangian funtion for the body with the effetive mass density tensor L= The equations of motion have the form µ d d has form 5

6 dp µ d d = 0 (9) d p = is the effetive density of the four-momentum, τ is the proper time. where: The effetive density of the four-momentum is an eternal physial quantity, whih desribes the relationship the tested objet with respet to the all surrounding bodies, not respet to the spaetime. dp For a hange of the effetive density of the four-momentum at proper time τ, i.e., in the ( ) µ d d equation (9), is responsible the term, whih desribes the distribution and motion of the all surrounding masses. The body motion does not depend on the inner properties of the spaetime, but the from presene and the motion of the all surrounding bodies and their distribution. The new quality of the understanding, kept in Mah s spirit, has been reahed. When the all surrounding bodies onsist only with the masses, i.e. when dp =0 and the effetive density of the four-momentum p is onstant along the world line. The left side of the equation (9) has a different, equivalent form dp In ase, when d = d d = d + d = 0 does not depends epliitly on τ, the equation (9) has form ( ) +Γ µ d d ( ) = 0 =0, then d (0) where: Γ ( ) = def µ + µ Inertial fores appears, when the body hanges its state of motion only with respet to the all surrounding bodies. This is the reason why E. Mah was led to make the attempt to eliminate spae as an ative ause in the system of mehanis. Aording to him, a material partile does not move in not aelerated motion relatively to spae, but relatively to the entre of all the other masses in the Universe; in this way the series of auses of mehanial phenomena was losed, in ontrast to the mehanis of Newton and Galileo []. 6

7 .7 The rotation buket with water problem When the buket of water put at the pole then, due the Earth s rotation, the water in a buket will hange his state of motion relative to the all surrounding bodies and will appear the entrifugal fore. The water surfae will adopt a paraboli shape. If a buket of water will be turned with the same angular veloity as the rotating Earth, but in the opposite diretion, then surfae should be flat, beause surfae of water will be at rest respet to all the surrounding bodies. What will happen if, instead buket of water, we put the Sagna interferometer? The eperiment an onfirm (or not) Berkeley and Mah point of view, that only rotation with respet to the all surrounding masses, have physial sense. The rotation of the body with respet to the spaetime loses its raison d'être and the spaetime eases to be the frame of referene..8 The medium as the referene frame In partiular ase, when the all surrounding masses are the masses, then the inertial fores will disappear d = 0. () The body with the mass density is in the rest or moves in a straight line with the onstant speed in respet to the all surrounding masses. The equation () determines the new referene frame the medium referene frame..9 The equation of motion in a weak field Suppose that, the surrounding masses and their distribution generates a weak gravitational field. The equation of motion (0) takes now the form (i =,, 3) i 00 i d + 00 () dt The equation () is different than Newton's equation of the motion for the gravity. It what urrently we onsider to be the inertial mass density, really is the sum of the mass density and the effetive mass density. The effetive mass density of the body is a results of interations between the body and the all surrounding bodies, during the hange state of motion. Note that in the equation () gravitational mass does not appear epliitly, so the Equivalene Priniple, underlying the GR, lost raison d'être. 3 Whih medium is orret? The idea of gravitational waves is a very attrative to many researhers. Although the GR survived so far all tests observation, diret detetion of the gravitational waves still remains missing part of the puzzle. It is believed that the gravitational waves an onvey important information about the early 7

8 Universe. However, despite strong irumstantial evidene for the eistene of these waves, so far, they have not been deteted diretly. This is ontrary to epetations many theorists therefore the very important question: Why is there no gravitational waves? is still open. The left side of the equations () and (5) desribes the two different media, respetively: the urved spae-time medium and the effetive medium, in whih the gravitational waves are propagated. Thus, there are two different ways desription of the propagation of these waves, as: a flutuations in the urvature of the spaetime or a flutuations in the effetive mass density []? Whih one is orret? Conlusion To desribe the gravitational phenomena applied a new mathematial model, whih merges the Minkowski spaetime and the mass density into the single idea. This desription is kept in the spirit of Mah, in opposite to the Newtonian gravity and GR. All phenomena of gravity are epressed in terms of the relationship between bodies and not between the body and surrounding the spaetime. Interation, whih appears during the hange state of motion, and whih propagates immediately, wakens our hope. Referenes []. Einstein, The Meaning of Relativity, Prineton University Press, published 9, p. 59. []. M. J. Kubiak, The Afrian Review of Physis, vol. 9, 0, pp