英语论文网

e have dx = g[dX + v dT], cdt = g[cdT + (v/c)dX], where g = [1 – (v/c)2]-1/2. It thus follows (18) that there is no time dilation although gdx’ = dX. Thus, K’ is not a physical space. This shows Tolman does not understand the need of a local time in Einstein’s theory (see section 4). Moreover, if Tolman’s calculation were valid, he actually showed that Einstein’s equivalence principle were invalid. In Einstein’s [8] analysis, the effects of an accelerated frame can be related to a gravitational potential F, which is a function of spatial variables in Newtonian theory as shown in eq. (3). But, all the metric elements of (19) are functions of time t’. Although Gx’t’t’ ¹ 0, the non-zero term in (21a) comes from gt’x’ but not from gt’t’ (since = 0 for m ¹ t’). Tolman simply ignored that Einstein’s later paper [1,8] confirms his 1911 analysis, and one has the relations, . + 2F¤c2, and ai » - ¶F¤¶xi (22) where F is the negative gravitational potential and a function of x’. Obviously, (19) is not consistent with equation (22). Thus, if Einstein’s equivalence principle is valid, metric (19) cannot be a physical space. Since Tolman believed in physical validity of arbitrary coordinates, he would also disregard that. (19) would imply the light speed in the x-direction to be –at’± c. In an attempt to overcome the deficiency of metric (19), in 1958 Fock [11] modified transformation (18) with x = x’ – at’2 , t = t‘ – at’x’/c2. (23) However, Fock does not seem to have any justification other than a desire of putting a new term into the metric element gtt so that its spatial derivative of related gravitational potential would generate the required acceleration as (22). Then, he obtained ds2 = (c2 - 2ax’ - a2t’2) dt’2 – dx’2 – dy’2 – dz’2 + a2(t’dx’ + x’dt’)2/c2. (24) The term 2ax’ seems serve the purpose, and metric (24) would be superficially compatible with relation (22). However, the equation of motion even for dx’/ds = 0 is very complicated as follows: = a c-2(1 – a2t’2/c2)-1[(1 – ax’/c2)2 – a2t’2/c2]-1. (25a) It is clear that (25a) is not exactly a uniform acceleration. Also, validity of (24) required two more inequalities as follows: 1 – a2t’2/c2 > 0; and (1 – ax’/c2)2 - a2t’2/c2 > 0. (25b) However, the second inequality cannot be justified in terms of physics, and can be traced back to the unjustified second transformation in (23). Thus (23) also does not lead to a physical space. Moreover, the velocity of light in the x’-direction would be >dx’/dt’ = {a2x’t’/c2 ± (c – ax’/c – a2t’2/c)}/(1 – a2t’2/c2). (26) But, the light bending experiment supports that a speed reduction under gravity. Since metric (24) is time-dependent, this also disagrees with observation. Moreover, metric (24) is in fundamental theoretical disagreement with Einstein’s earlier and also subsequent analysis [1,16]. In short, metric (24) still does not represent a physical space. Nevertheless, Fock [11] still believed that the problem of time-dependence could be resolved within the framework of the speculated metric (4). Fock [11] proposed the following mathematically ingenious transformation, x