THE TWIN PARADOX A RELATIVISTIC DOMAIN RESOLUTION

Transcription

1 THE TWIN PARADOX A RELATIVISTIC DOMAIN RESOLUTION Peter G.Bass P.G.Bass January 0

2 ABSTRACT This short paper shows that the so alled "Twin Paradox" of Speial Relativity, is in fat not a paradox at all, and is easily resolvable using the Relativisti Domain mathematial formulation of the Speial Theory. P.G.Bass i

3 CONTENTS.0 Introdution..0 The Twin Paradox Resolution.. The Existene Veloity of Both Twins in Pseudo-Eulidean Spae-Time.. Assessment of Spatial Veloities..3 Assessment of Travel Time. 3.0 Assessment of Referene Frame Symmetry. 4.0 Conlusion. Referenes. P.G.Bass ii

4 .0 Introdution. The "Twin Paradox" is a thought experiment in Speial Relativity, in whih the ages of twins are ompared after one has made a high speed journey over a long period of time, while the other has remained stationary. The proposed paradox arises beause eah twin sees the other as travelling, and so should apparently, due to the effets of relativisti time dilatation, see the other as having aged more slowly than himself. Ever sine Einstein's demonstration of the presene of time dilatation in the Speial Theory, there has been a signifiant amount of disussion on the "Twin Paradox". Some protagonists maintaining that it represents an unresolved issue, while others hold the belief that it is not a paradox at all, primarily beause of the lak of symmetry in the inertial frames of referene of eah twin. Aordingly, there have been a onsiderable number of attempts, some of whih are ontinuing, [], to demonstrate this, but, to date, none have proved to be entirely satisfatory, []. It is believed that this is beause suh attempts have not fully onsidered the temporal aspets of the problem, only the spatial. The analysis in the remainder of this short paper will approah the problem using the Relativisti Domain formulation of the Speial Theory, as presented in [3], thereby giving the temporal issues equal prominene with the spatial..0 The Twin Paradox Resolution. In the ensuing analysis, eah twin assumes the role of observer of the other in their separate frames of referene. Thus the stationary twin is designated observer O S, while the travelling twin, observer O T.. The Existene Veloity of Both Twins in Pseudo-Eulidean Spae-Time. The Existene Veloity, as defined in [3], of eah twin in the referene frame of O S is shown below in Fig... (i) The outward journey of O T. V V v V V O S O T (ii) The inward journey of O T. V V V V v O S O T Fig.. - The Existene Veloities of O S and O T in Pseudo-Eulidean Spae-Time. P.G.Bass

5 . Assessment of the Spatial Veloity of Eah Twin by the Other. It is important that eah observer orretly determines the spatial veloity of the other in his own frame of referene. (i) The average spatial veloity of O T as determined by O S is simply Outward journey Inward journey dx v (.) dt dx dt v (.) (ii) The average spatial veloity of O S as determined by O T is, using the Lorentz transformations between referene frames. Outward journey and therefore Inward journey dx vdt vdt d x (.3) v v dt dx v dt d t (.4) v v dx v (.5) dt dx + vdt vdt d x (.6) v v and therefore dt + dx v dt d t (.6) v v dx + v dt Hene eah twin orretly assesses the spatial veloity magitude of the other. (.7) It is important to note that (.4) and (.6) is a measure of the length of a unit of time in the referene frame of O T, and not the journey travel time. P.G.Bass

6 .3 Assessment of Travel Time, (Age Differene). Beause this is only a thought experiment, the assessment of the temporal omponent of Existene Veloity of eah twin by the other, may be inorporated as a realisable measurement. (i) The travel time of O T as determined by O S. The travel time of the round trip as measured by O S is t. The temporal veloity of O T, for both outward and inward journeys is, from Fig.. given by (.8) Therefore the temporal distane travelled by O T, as determined by O S is x t t 0 and onsequently, the amount of time that has passed for O T as assessed by O S is (.9) x (ii) The travel time of O S as determined by O T. 0 t t (.0) The time of the round trip as measured by O T is given by (.0), i.e. t. The temporal veloity of O S as measured by O T is Thus the temporal distane travelled by O S, as determined by O T is x0 t (.) (.) and therefore the amount of time that has passed for O S as assessed by O T is from (.8) and (.) x 0 t t (.3) Therefore, by (.0), O S assesses that O T has aged less than himself by the amount ( t - t ), and onversely, by (.3), O T assesses that O S has aged more than himself by the same amount. P.G.Bass 3

7 3.0 Assessment of Symmetry. The degree of symmetry between the referene frames of O S and O T an be determined from the above results as follows. (i) Spatial Distane. (a) The distane moved by O S in his own frame of referene is zero. (b) () (d) The distane moved by O T, as assessed by O S, using a unit measuring rod, (all measurements taken with the rod stationary), is D. The distane moved by O T in his own frame of referene is zero. The apparent distane moved by O S, as assessed by O T using a unit measuring rod, (all measurements taken with the rod moving with a veloity in Pseudo- Eulidean spae of v), is ( v ) D by virtue of the Lorentz-Fitzgerald ontration of the measuring rod. (b) and (d) show that the spatial dimensions of the two referene frames are not symmetri. (ii) Temporal Distane. (a) The temporal distane travelled by O S in his own frame of referene is t. (b) The temporal distane travelled by O T, as measured by O S is given by (.9),. i.e. ( v ) t () The temporal distane travelled by O S as measured by O T is given by (.) and is, with (.8) and (.0) inorporated, ( v ) t. (d) The temporal distane moved by O T as measured in his own frame of referene is t ( v ) t. These results show that the temporal dimensions of the two referene frames are not symmetri. This non-symmetry of distane between the two referene frames is the result of the rotation of the Existene Veloity vetor of O T. (iii) (iv) Time. The passage of time is not a symmetri parameter between the two referene frames as exemplified by the time dilatation effet, and onfirmed by the results of this analysis, albeit eah twin an orretly identify the age differene. Veloity. Setion. shows that spatial veloities in the two referene frames are symmetri, (in this ase diametrially symmetri, same magnitude, different diretions). Hene the veloity of light is the same in both frames. This symmetry arises beause the nonsymmetry of spatial distane is offset by the non-symmetry in time. P.G.Bass 4

8 The temporal veloities of O S and O T are and ( ) v respetively and therefore not symmetri. This non-symmetry exists beause the non-symmetries in temporal distane and time are onurrent as shown by (ii)(d) above. 4.0 Conlusions. The use of the Relativisti Domain mathematial formulation of the Speial Theory, has enabled this paper to easily show that the "Twin Paradox" is in fat not a paradoxial problem at all, as eah twin an orretly assess the age differene as a result of the journey of one of them. Albeit this has been ahieved by allowing eah observer to assess the temporal veloity of the other, a measurement not pratiable in real world problems, and only permitted here beause of the nature of the experiment. The age differene orretly identified by eah twin is essentially the result of the nonsymmetry of temporal veloity, while importantly, the veloity of light is the same in both referene frames beause of the symmetry of spatial veloity between them. Also, it is noted that the analysis here does not require the neessity of onsidering the aeleration involved in the motion, nor the effet of gravity. Finally, it has also been shown that the nature of symmetry between the two referene frames is not just a simple overall one, but depends upon the dimensions ontained therein and the parameter involved. Referenes. [] NPA, Twin Paradox Report, Online. [] Wikipedia, Twin Paradox, Online. [3] P.G.Bass, The Speial Theory of Relativity - A Classial Approah, P.G.Bass 5