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IX. Inertia! Frame given by a Hyperbolic Space-time. 



To the Editors of the Philosophical Magazine. 

 Sirs, — 



IT is interesting to modify de Sitter's inertial frame to a 

 hyperbolic form. Referring to Professor Eddington's 

 valuable 'Report' on Relativity, re-write his (51.1) thus 



dJ = — W\d& + sinh 2 e(d<9 2 + sin 2 '0d<£ 2 ) \ +.cosh 2 ®^ 2 . . (51.1) 



His (51 .2) and (51.3) become 



isinh B= sin f sin go, tan (t/'R)= cos f tan a>, . . (51.2) 

 ds 2 =:K 2 (dco 2 +*'m 2 cD(d£ 2 + sin 2 £(i0 2 -+ sin 2 &ty 2 ))), (51.3) 



the only change here being in the sign of R 2 . His (52.2) 

 changes in the same way, or 



ds 



2_ 



-dr* r 2 ,, A9 . . 9/I7 -„ x . dt* 



(l-er 2 y 



— -^ OM» + rin f fcfy')+- ~~ , .(52.2) 



where e=l/R 2 . [In the work below I alter the scale of 

 this three-dimensional map, putting ?'=tanh(8), e = l in 

 place of r=R tanh ©, e=l/R 2 .] 



The effects of the modification are first that there is no 

 time-barrier ; to a fixed observer the converse of what 

 occurs in de Sitter's space will happen — things will seem to 

 move uncannily fast in the distant parts of the universe and 

 distant spiral nebulae should show a spurious tendency to 

 approach by a spectrum shift towards the violet. 



The relative motion of two particles undisturbed by 

 gravitation is excessively simple but, to my mind at least, 

 most extraordinary and unexpected. Let one of the par- 

 ticles be taken as fixed at the origin and let the motion of 

 the other be mapped after the manner of (52.2). It will 

 not move in a straight line but in the ellipse 



r 2 r 2 



— + — = 1 



a 2 + 6 2 ' 



for which t/~R is the eccentric angle. Thus all such undis- 

 turbed particles retrace their sinusoidal motions for ever 

 and all have the common period 27rl\ (velocity of light = 1), 

 Moreover, every observer is the centre of all the elliptic 

 orbits. 



In the last paragraph t is coordinate time, but there is no 



