Motion in a Non-Inertial Frae of Reference vs. Motion in the Gravitoanetical Field Mirosław J. Kubiak Zespół Szkół Technicznych, Grudziądz, Poland We atheatically proved that the inertial forces, which appears in a noninertial frae of reference, such as acceleratin and rotatin reference frae, are equivalent to the real forces which appears, when the body oves in the ravitoanetical field. We will reind the rotatin bucket with water proble with the new proposal of the solution. ravitation, ravitoanetis, four-vector field of velocity, inertial forces INTRODUCTION Let's replace the scalar potential of a ravitational (G field, ϕ(r, t, by the V (r, t, where V (r, t = - ϕ(r, t. Let s nae the V (r, t as the scalar field of the square of the velocity. Let's replace the vectorial potential of the ravitoanetis (GM field A (r, t by the V (r, t, where V (r, t = A (r, t. Let s nae the V (r, t as the vectorial field of the velocity []. Let's replace of the G and GM four-potential A µ = (ϕ/c, A by the four-vector field of the velocity (V µ, which we will define in the for V µ def V = c, V where: c speed of propaation of field (equal to, by General Theory of Relativity, the speed of liht c. The (V µ has diension [/s], fro here the nae - the four-vector field of the velocity []. The nonrelativistic Laranian for the body with ass ovin in the external scalar field of the square of velocity (V in an inertial frae is v0 L 0 = + V ( and the correspondin equation of otion = (V ( In this section the suffix 0 denotes quantities pertainin to an inertial frae.
Let us now consider what the equations of otion will be in a non-inertial frae of reference. The basis of the solution of this proble is aain the principle of least action, whose validity does not depend on the frae of reference chosen. Larane's equations d L L = v r (3 are likewise valid, but the Laranian is no loner of the for ( and to derive it we ust carry out the necessary transforation of the L 0. This transforation we will realize in two steps []. STEP. ACCELERATED TRANSLATIONAL MOTION OF THE FRAME OF REFERENCE Let us first consider a frae of reference K' which oves with a translational velocity V(t relative to the inertial frae K 0. The velocities v 0 and v' of a body in the fraes K 0 and K' respectively are related by Substitution of this in ( ives the Laranian in the frae K': v 0 = v' + V(t (4 v ' V (t L ' = + v'v(t + + V (5 Now V (t is a iven function of tie, and can be written as the total derivative with respect to t of soe other function, the third ter in Laranian function L' can therefore be oitt'ed. Next, v' = dr'/, where r' is the radius vector of the body in the frae K' []. Hence dr' d dv V(t v' = V = ( Vr' r' (6 Substitutin in the Laranian and aain oittin the total tie derivative, and we have finally v ' L ' = a(t r' + V (7 where a(t = dv/ is the translational acceleration of the frae K'. The Larane's equation of otion derived fro (7 is ' = (V a(t (8 Thus an accelerated translational otion of a frae of reference is equivalent, as reards its effect on the equations of otion of a body, to the application of a unifor field of force equal to the ass of the body ultiplied by the acceleration a = dv/, in the direction opposite to this acceleration []. STEP A. MOTION IN THE ACCELERATED (V µ FIELD Let us consider the Laranian for the body with ass ovin in the scalar field of the square of velocity (V and in the vectorial field of velocity of the V (t
v L = + vv (t + V (9 Calculations ive equation of otion = (V a (t (0 where a = V (t/ t is the vectorial field of the acceleration. Coparin two equations (8 and (0 we can see, that they are equal, if and only if, when the a = a. STEP. ROTATION OF THE FRAME OF REFERENCE Let us now brin in a further frae of reference K, whose oriin coincides with that of K', but which rotates relative to K' with anular velocity ω(t. Thus K executes both a translational and a rotational otion relative to the inertial frae K 0 []. The velocity v' of the body relative to K' is coposed of its velocity v relative to K and the velocity ω r of its rotation with K: v = v + ω r (since the radius vectors r and r' in the fraes K and K' coincide. Substitutin this in the Laranian (7, we obtain v L = + v ( ω r + ( ω r a r + V ( This is the eneral for of the Laranian of a body in an arbitrary, not necessarily inertial, frae of reference. The equation of otion has for []. V dω = + r + r ( v ω + ( ω ( r ω ( We see that the inertia forces due to the rotation of the frae consist of three ters: dω. the force r is due to the non-unifority of the rotation,. the force ( v ω is called the Coriolis force, ω r ω is called the centrifual force []. 3. the force ( ( STEP A. MOTION OF THE BODY IN THE ROTATING (V µ FIELD Let us consider the Laranian function for the body with ass ovin in the scalar field of the square of velocity (V (r and in the vectorial field of the V (r, t v V L = + vv + + V (3 Calculations ive the equation of otion [3] We oitted the ter relatin to the acceleration a r, concentratin on the rotation only. 3
V dω = + r + r ( v ω + ( ω ( r ω (4 where: ω is the intensity of GM field or the vectorial field of the rotation []. Coparin two equations ( and (4 we can see, that they are equal, if and only if, when the ω = ω. We atheatically proved that the inertial forces, which appears in a non-inertial frae of reference, such as acceleratin and rotatin reference frae, are equivalent to the real forces which appear, when the body oves in the GM field (the vectorial field of velocity. Does this sinify that applyin the GM field (the vectorial field of velocity we can eliinate a inertial forces? It is the very iportant question. Now we will reind below the rotatin bucket with water proble with the new proposal of the solution. THE ROTATING BUCKET WITH WATER PROBLEM Berkeley and Mach criticized the Newtonian interpretation of the experient of the rotatin bucket with water [3]. When the bucket rotates with respect to the fixed star sphere (FSS (all bodies in the Universe, then the surface of water has a paraboloidal shape. Instead of rotatin the bucket, assue we could turn, by the sae eans, the FSS so that the relative rotation is the sae. Newton thouht that if we rotated the FSS, then because the otion of the water is described with respect to the absolute space, its surface would be flat. A contrary point of view, that is, that only rotation with respect to the FSS ives curvature of the water surface in the bucket, was proposed by Berkeley and Mach. Accordin to the, the absolute space is unobservable. Mach pointed out that it does not atter if the Earth is rotatin and the FSS is at rest, or stationary Earth is surrounded by the rotatin FSS. His idea is based on an epirical fact: two easureents of the Earth s anular velocity, astronoical (with respect to the FSS and dynaic (by eans of Foucault s pendulu experient, ive the sae results (in the liits of the experiental errors. THE ROTATING LIQUID MIRROR WITH MERCURY INSTEAD THE BUCKET WITH WATER The rotatin liquid irror (LM [4] with ercury, instead the bucket with water, could confirs (or not the Berkeley and Mach point of view that the rotation (only with respect to the FSS ives curvature of the surface of the ercury. If we were to take the LM of the ercury, with the utost care, to the Earth s pole, we would find that the surface of the ercury assues a paraboloidal shape, even when the LM is at the rest relative to the Earth (the Earth with LM is rotatin relative to the FSS. Accordin to the equation ( the heiht of ercury h(r, in the coordinate syste uniforly rotatin with respect to the stationary FSS with ω anular velocity, has the for [5] h(r = h(0 + ( ωr (5a E where: E 9,8 /s is the G acceleration, h(0 is the heiht of the ercury at r = 0, r is a radius of irror. The surface of the ercury is parabolic in its dependence upon the radius of irror. 4
Accordin to the equation (4 the heiht of ercury h(r, in constant hooenous vectorial field of rotation ω, has the for h(r ( ω r = h(0 + (5b E If we apply the otion to the ercury with the sae anular velocity and in the sae direction that the FSS rotates, then the surface of ercury would reain flat, because ω - ω = 0 and h(r = h(0. CONCLUSIONS In this paper we atheatically proved that the inertial forces, which appears in a non-inertial frae of reference, such as acceleratin and rotatin reference frae, are equivalent to the real forces which appears, when the body oves in the GM field (the vectorial field of velocity. Does this sinify that applyin the GM field we can eliinate a inertial forces? It is the very iportant question. The new experient with the liquid irror with ercury instead the bucket with water could confirs (or not the Berkeley and Mach point of view that the rotation (only with respect to the FSS ives curvature of the surface of the ercury. REFERENCES. M. J. Kubiak, Consequences of Usin the Four-Vector Field of Velocity in Gravitation and Gravitoanetis, http://vixra.or/abs/0.0036, Oct 0.. L. D. Landau, E. M. Lifshitz, Mechanics, 3 th ed., Butterworth-Heineann, p. 6, 000. 3. M. J. Kubiak, The rotatin bucket with water proble, Physics Essays, vol. 6, No. 4, 993. 4. The International Liquid Mirror Telescope Project, http://www.aeos.ul.ac.be/lmt/, May 0. 5. J. M. Knudsen, P. G. Hjorth, Eleents of Newtonian Mechanics, Spriner, 000. 5