The Concept of the Effective Mass Tensor in GR. The Equation of Motion

Transcription

1 The Concept of the Effective Mass Tensor in GR The Equation of Motion Mirosław J. Kubiak Zespół Szkół Technicznych, Gruziąz, Polan Abstract: In the papers [, ] we presente the concept of the effective ass tensor (EMT) in General Relativity (GR). Accoring to this concept uner the influence of the gravitational fiel the ass tensor becoes the EMT. The concept of the EMT is a new physical interpretation of GR, where the curvature of space-tie has been replace by the EMT. In this paper we consier again the concept of the EMT in the GR but in the aspect of the equation of otion. keywors: the effective ass tensor, the equation of otion, Mach s principle I. Introuction In the papers [, ] we presente the concept of the effective ass tensor (EMT) in General Relativity (GR). Accoring to this concept uner the influence of the gravitational fiel the ass tensor becoes the EMT (see Table I). The concept of the EMT is a new physical interpretation of GR, where the curvature of space-tie has been replace by the EMT. In this paper we again consier the concept of the EMT in the but in the aspect of the equation of otion. In the Table I we copare a few physical features concerning of the space-tie curvature with the physical features of the EMT which were iscusse in this paper. As we know fro the [, ] the etric tensor g we can express by the EMT g () where: is the ass of the boy, the space-tie coponents,,,, 3. Therefore the etric s ( g ) s ( ) ()

2 s g g s where: ( ) an ( ) II. The equation of otion Let us consier the Lagrangian function for the free boy in the curve space, which is oving with the sall velocity ( << c), where c is the spee of the light, τ is the proper tie.. L g (3) If we replace in the eq. (3) the etric tensor g with the EMT (see eq. ()) then we have L (4) The equation of otion for the Lagrangian function (4) have the for (see Appenix) (5) where the ter we will call the oifie Christoffel sybols of the secon kin an α α α α (6) In the weak gravitational fiel we can ecopose of the EMT of the boy to the siple for:, where: η iag(,,, ) we will call the ass ten- sor, η is the Minkowski tensor, h << is a sall EMT perturbation [] (see also to Table I). Note that in the absence of the gravitational fiel the EMT becoes the ass tensor. The escribes the anisotropy of the EMT. For the ass tensor the. The oifie Christoffel sybols (6) (with accuracy to first orer) have the for i i δ (7) (coponents i an are the Roan inices to enote spatial coponents: i,,, 3) an siilarly Now the equation of the otion (5) have the for ( ) i ik δ k k (8)

3 i ( k k ) i ik c δ δ c (9) where we oitte the ter. Now we can interpret this equation in the Newtonian language. The secon right ter in the eq. (9) is velocity-epenent an is associate with the rotation an the Coriolis acceleration. Let s assue that the boy is not rotate an. Motion of the boy in a weak gravitational fiel is escribe by the equation k k i i c δ () Accoring to the eq. () we can say that the graient of the tie coponent of the EMT i eterines the acceleration EMT is an anisotropic an of the teste boy in the gravitational fiel. If the (the ) then the teste boy is oving with an acceleration i. But if the (the EMT is an isotropic an ) then the acceleration of the boy an the test boy ove takes place at a constant velocity. Eq. () eter- i ines the first Newton s law of otion in the EMT fraework. Of course we ust reeber that our consieration we use for a weak gravitational fiel in the Newtonian approxiation. The ore generally conition in eq. (5) eterines the first Newton s law of otion in EMT fraework an is ore precise. In the classical echanics the Newtonian equation of otion have the for i V () where: V is the Newtonian gravitational potential of the source. Both eq. () an () are equivalent if an only if the tie coponent of the EMT takes the for V GM () c c r We can see that the concept of the EMT correctly escribes gravitational phenoena in the Newton's law of gravity. Let us consier the Lagrangian function (eq. (4)) with the Schwarzschil EMT (the EMT in the Schwarzschil etric) The EMT is isotropic if an only if his properties o not epen on the irections in the space-tie []. So far, the concept of the ass isotropy was associate only with the 3-iensional space but not with the 4- iensional space-tie. It is a new paraig. 3

4 Schwarzschil GM c r GM c r r r sin θ (3) Well known calculations satisfy the classical tests of GR for exaple: the perihelion shift, the eflection of light by the Sun an the gravitational reshift, but their physical interpretation is ifferent. The concept of the EMT correctly escribes gravitational phenoena known fro GR but these physical phenoena are not generate by the curvature of the space-tie, but by the effective ass tensor of the boy. Let s copare a few physical features concerning of the space-tie curvature with the physical features of the EMT iscusse in this paper. Results of coparison are presente in Table I below. Table I. The space-tie curvature vs. the EMT. Space-tie curvature The EMT The etric tensor g The effective ass tensor g The etric ( g ) g s The etric s ( ) Decoposition of the etric tensor in the weak gravitational fiel g η h Decoposition of the EMT tensor in the weak gravitational fiel where: η iag(-,,, ) is the Minkowski tensor, h << is a sall perturbation. where: iag(-,,, ) η is the ass tensor, << is a sall EMT perturbation. h Lagrangian: L g Lagrangian: L 4

5 The equation of otion ( ): where the ter is the Christoffel sybols an g α g α g α g α The equation of otion: where the ter we will call the oifie Christoffel sybols an α α α α Equation of otion in the Newtonian liit Equation of otion in the Newtonian liit r i c δ h V r i c δ V III. The Foucault's penulu There are two ifferent ways of easuring the Earth s rotation about its polar axis. The first one is astronoical etho where we can eterine Earth's rotation with respect to the backgroun of istant stars. The secon etho is ynaically etho where Earth's rotation we can eterine by eans of Foucault's penulu or the gyroscope. Both ethos have the sae results. The first etho escribes Earth s rotation with respect to the fixe stars. The secon one with respect to the absolute space. Is it iportant coincience or not? Accoring to E. Mach this coincience is not trivial an only in equation (8) changes the plane of Foucault's penulu. the ass rotation ( ) k k IV. Mach s Principle Eq. (5) we can rewrite in a slightly ifferent for (4) where the ter forula we will call the oifie Christoffel sybols of the first kin expresse by the 5 Now the otion of the boy in the gravitational fiel is escribe by two interesting expressions: an. The first ter escribes the four force, where: is the EMT, is the four acceleration (see Appenix). Diension of this ter is [N kg /s ]. The (5)

6 secon ter escribes the four-graient of the EMT. Diension of this ter is [kg/]. While the ter escribes the four-graient of the EMT an four-velocities of the effective asses. Diension of this ter is [N ]. Accoring to eq. (4) we can say that the effective ass of the boy is not an intrinsic property but is a result of interactions between this boy an the effective ass other boies in the Universe. This is Mach s Principle in a new soun. V. Conclusion In this paper we consiere the concept of the EMT in the aspect of the equation of the otion. Accoring to this concept uner the influence of the gravitational fiel the ass tensor becoes the EMT. The concept of the EMT is a new physical interpretation of GR, where the curvature of space-tie has been replace by the EMT. Analyzing the Lagrangian function (4) an the equation of otion (5) we see that: in a weak gravitational fiel the Lagrangian function an the equations of otion for the boy with the ass oving in the space-tie curvature with the etric tensor g are the sae like the Lagrangian function an the equations of otion for the boy oving with the EMT in the flat Minkowski space-tie. Both escriptions are equivalent (see Table I). In the Newtonian liit the equation of otion (5) gives the equation () well known fro classical echanics. Notice that the otion of the boies we escribe relative to other boies with the effective or ass, but not to the respect to the assless reference fraes. The concept of the EMT is a very attractive because the equation of the otion (5) inclues full inforation about all fiels (in this case a weak gravitational fiel) surrouning the boy without their exact analysis. The EMT can be isotropic or anisotropic. Us we known fro the soli-state physics the EMT can be positive or negative. Is a negative effective ass can exist in a Universe? It is a very iportant question. The concept of the EMT offers Mach s Principle in a new soun: the effective ass of the boy is not an intrinsic property but is a result of interactions between this boy an the effective ass other boies in the Universe. We believe that the concept of the EMT will help better unerstan a fascinating gravitational phenoena. Appenix Let us consier the Lagrangian function with the EMT ( ) ( x ) x L (A) 6

7 The Euler Lagrangian equations has the for τ ( ) (A) After well-known calculation we get the equation of otion (5). If EMT oes not epens on coorinate x then τ ( ) (A3) which iplies that is a constant. But, so we have that the fouroentu p ( ) ( ) is constant. Thus, we obtaine a very iportant relationship between four-oentu p an EMT an p (A4) Note that this relationship is soething ifferent than in GR, where ass is a scalar. Using eq. (A4) we can rewrite eq. (4) in a ore general for p (A5) where: p is the four-oentu. Differentiating the four-oentu in eq. (A5) with respect to we obtain (A6) Although the ass oes not exist explicit in eq. (5) but plays a very iportant role in the eq. (A6). If we assue that then eq. (A6) becoes the eq. (4). References []. M. J. Kubiak, The Concept of the Effective Mass Tensor in the General Relativity, []. M. J. Kubiak, The Concept of the Effective Mass Tensor in GR. Clocks an Ros, 7