Test of General Relativity Theory by Investigating the Conservation of Energy in a Relativistic Free Fall in the Uniform Gravitational Field

Transcription

1 Test of General Relatiity Theory by Inestigating the Conseration of Energy in a Relatiisti Free Fall in the Uniform Graitational Field By Jarosla Hyneek 1 Abstrat: This paper inestigates the General Relatiity Theory (GRT) by studying the relatiisti free fall of a small test body in a uniform graitational field. The paper ompares the preditions of energy loss, perhaps by radiation, in a free fall obtained from the GRT and from the Metri Theory of Graity (MTG). It is found that the graitational mass dependene on eloity in GRT is not orret, beause it predits a negatie loss of energy while the MTG predits orretly a positie loss. Introdution: The theories desribing the free fall are well understood in both; the GRT and the MTG. In the GRT the inertial mass and the graitational mass are assumed idential with idential dependenies on eloity. In the MTG, on the other hand, the graitational mass depends on eloity differently than the inertial mass [1, ]. It is thus simple for both theories to derie equations desribing the free fall eloity and from that the energy loss of a small test body that falls in a uniform graitational field. Theories: In the GRT the relation between the eloity and time is somewhat more ompliated but an be easily deried as follows: d m m = g (1) where m is the rest mass and the speed of light in a auum. The left hand side of Eq.1 is the relatiisti formula for the inertial fore and the right hand side is the formula for the graitational fore that inludes the graitational mass dependene on eloity. The formula in Eq.1 an be rearranged and simplified resulting in the following relation for the small body aeleration: d g(1 / ) = () The energy loss will be alulated by omparing the potential energy that is obtained by lifting the small test body ery slowly in the uniform field by a distane z to the energy of the falling body. The test body potential energy is simply expressed as follows: E= mg z () This relation will be kept as referene energy een if the test body may moe fast. For the atual falling body energy the inremental energy gain by a fall an be expressed in terms of the eloity and the graitational fore F as follows: d de = F = g m d d g () 1 jhyneek@netsape.net ; 1 Isetex, In. 1

2 After rearrangements and integration the expression for the energy as a funtion of eloity beomes: E = m (5) This result is expeted and it is a nie onfirmation of methodology used in Eq.1. In the next step we will ealuate the energy differene E gien by Eq. and Eq.5. Howeer, for the oneniene of alulations it will be useful to first ealuate the time deriatie of this differene. d E = mg m ) In this formula the ariable in the first term was substituted for the time deriatie of z. By substituting for the aeleration from Eq. the result beomes: d d E 1 1 = mg 1 mg By integrating this result in the eloity domain, similarly as it was shown in Eq., we obtain the expression for the total energy differene loss during the fall: (6) (7) m () This is a ery strange result. It seems that the falling body is gaining some additional energy on top of the energy that is predited from the free fall by Eq.5. This is not reasonable and it is pointing to a problem that exists in the GRT for a long time. The graitational mass annot depend on eloity the same way as the inertial mass. This problem will beome lear from the result presented next. The similar expression introdued in Eq.1 is used, but with the graitational mass depending on eloity as follows [1, ] : This leads to the following formula: mg m 1 / = (9) d m = g m (1) After rearrangements the formula is simplified with the result as follows: d g(1 / ) = (11) Following the same proedure as aboe for the GRT ase the differential of energy will be:

3 de = g m d ) This beomes, after integration, idential to formula in Eq.5. Both theories, the GRT and MTG, thus gie the same expression for the energy, whih is expeted and onfirms one more that the alulating proedure is orret. For the energy loss the same proedure is also followed with the result: d m d 1 = mg = mg mg ( 1 ) g 1 1 This is a similar result as in Eq.7 exept that the energy loss differential is now positie as it should be. This onfirms the orretness of the graitational mass dependene on eloity and therefore disproes the alidity of GRT. The energy loss is likely due to the graitational radiation, beause the falling body is aelerated. By integrating the result from Eq.1 the energy loss is equal to: m The result in Eg.1 is deried again in Appendix with more alulation details: For a better understanding of the magnitude of radiated energy the graph of the energy loss as a funtion of the fall time for a mass of 1kg is shown in FIG.1. The relatiisti energy onseration test for a Free Falling body in uniform g field g = m m = s t :=,.1.. nt( t) := g t mo := 1. kg s Ε ( t) := mo g t (1) (1) (1) FIG.1 the dependeny of energy loss due to radiation for a 1kg test body free falling in a uniform graitational field equal to Earth s graity. The loss for a se fall is about 1. miro Joules.

4 It may sometimes be useful to express the energy loss in terms of the fallen distane instead of the eloity. This an be deried starting from Eq.11 as follows: d = g dz dz ( 1 ) (15) After integration and rearrangements the relation between the fallen distane and the eloity beomes equal to: g z 1 1+ = (16) ) By substituting this result into the formula for the energy differene and onsidering the mass dependene on eloity as gien in Eq.9 the result beomes as follows: E= mg z+ m m 1+ g z (17) In this equation the rest mass equialent energy was added to the referene energy to be onsistent with the rest energy of the falling body. It is neessary that for the zero distane the energy differene and the eloity of falling body are both zero. By expanding the result from Eq.17 into a power series and negleting higher order terms the energy loss will be as follows: g z m m (1) The graph of this dependeny is shown for oneniene in FIG.. MTG energy onseration test in a Free Fall in terms of the fallen distane z g m mo g z = s z := m,. m.. 1 m mo := 1. kg ( z) := ( z) z Fig. the dependeny of energy loss for a 1kg test body free falling in a uniform graitational field.

5 Conlusions: The paper deried simple expressions for the energy loss during the small body free fall in a uniform graitational field. It was shown that the loss deried aording to the GRT is negatie. This is unaeptable and this fat thus disproes the alidity of GRT. This problem has its root ause in the idential dependeny of inertial mass and graitational mass on eloity. When the orret dependeny of graitational mass on eloity, as deried in the MTG, is used the orret positie energy loss is alulated. This result has fatal onsequenes for the GRT, beause unquestionably proes its inorretness. This finding thus has a signifiant impat on all the theories based on the GRT suh as the Big Bang and similar ridiulous models of the Unierse. The author hopes that the main stream relatiists finally reognize this problem and abandon the GRT with all its ridiulous laims of existene of Blak Holes, Eent Horizons, and the Big Bang Unierse with its aelerating expansion to infinity from nothing. Referenes: [1] [] Appendix: From Eq. we hae for the time deriatie of the referene energy the following: de dz = mg = mg (A1) The referene energy time deriatie should remain unhanged for small eloities as well as for large eloities. Howeer, for the energy deriatie of the falling body we must inlude the mass dependene on eloity from Eq.9. This results in the following expression for the deriatie of ΔE. d d = mg d Using the formula from Eq.11 and substituting it into Eq.A, the result beomes as follows: d E m m = d After integration the energy differene is found to be: ) ( 1 ) (A) (A) md md m m m 1 (A) E = = + ) ) ( ) This result an be expanded into a power series leading to the following final expression: ( ) m E... m = (A5) 5