Space-Time Models with (2+1) dimensions - (a) traditional model (b) with elimination of the past 

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Context 1

... on Space-time Topology The two rules stated above also have an influence on the type of geometry and topology that may be appropriate for a model of our world. The traditional view of SRT and GRT is that the space-time of our world has (3+1) 3 dimensions, which allows for various types of topological spaces, such as, for example, a 3-sphere, S . Leaving out 1 space dimension, the 2 space dimensions + 1 time dimension may be graphically represented as shown on the left hand side of Fig. 1. The right hand side of Fig. 1 shows the situation when the above two rules for the "Process of World State Progression" are applied. Instead of a topological space that is continuously expanding in the direction of time, we have a series of "time-slices". Rule2 says that there always exists only a single time-slice. The old time-slice is discarded when a new one is generated. If, however, the extension of the world in the time dimension is always equal to 0, one may question whether this should be viewed as a dimension at all; at the least, the time dimension has quite different characteristics than the space dimensions. One possible argument for keeping a time dimension is that it makes sense to label the series of time slices by unique time coordinates. However, if this is the justification for the assumption of a non-zero extension of the time dimension and Rule2 is assumed to hold, it should be made clear that the state of the world of past time slices is empty. When "time dimension" and "time coordinates" are discussed within this paper, it must be clarified that this always refers to the time represented by the time coordinates of the world's space-time, as opposed to the "proper time" known from SRT and GRT. Inclusion of the proper time in these discussions is outside the scope of this paper because it would not change the major conclusions of the paper (Note 1). Although the above described rules for the evolution of the world seem to be in accordance with many implicit assumptions in various areas of physics, some alternatives are discussed in this section . 5.1 Old states are not discarded - the past remains in existence Keeping past world states would seem to be a prerequisite for any time travel function. Let us assume that the past is retained in the form of a series of snapshots similar to a sequence of pictures contained in a movie. The follow-up question becomes this: what is the role and purpose of these world state snapshots with respect to the evolution of the world? Specifically, when and how may these snapshots become activated to play a role in the world's evolution? If the snapshots would never be used, their purpose (and existence) are highly questionable. Assuming that the snapshots are maintained just to enable time travel would be rather silly. Disregarding their oddness, there are two main alternatives recognized by the author for the way in which these potential past snapshots may be activated or used: 1. The snapshots remain permanently active, which means they are a permanent source for interpretations by the physics-interpreter. The continuous generation of new worlds at all stages of history would be the result. This is similar to the new worlds occurring in Everett's "many-worlds interpretation" of quantum theory (see Everett (1957)). In fact, such a scheme might be defined as an extension of Everett's many-worlds interpretation. 2. A modification of a past snapshot (e.g., as a consequence of a time travel function) automatically starts an update of the world by replacing all the snapshots that are younger than the modified snapshot up to the youngest (i.e., the present) state. In other words, changing the past would automatically and instantly result in a replacement of the present. As intended, these two alternatives for keeping the past would enable time travel (forgetting for the moment about some other physical problems). Notice, however, that both alternatives for utilizing the past do not result in the time travel paradox. With the "many-worlds process" (alternative 1), the non-existence of Tom Meyer would occur only in newly generated parallel worlds, which would not be a problem for the "old" worlds. With the "automatic update of the present" (alternative 2), the paradox is avoided because the contradicting statements occurring in the example described in the Introduction are now produced in sequence. Facts that would contradict when assumed concurrently may cause no problem when they occur in a certain sequence. The sequence would now be as follows: 1. Year 2005: Tom Meyer, a son of Tim Meyer, is born. 2. Year 2500: John Meyer, a successor of Tom Meyer, lives. 3. Year 2000, resumed: John Meyer performs actions that prevent the birth of Tom Meyer. 4. Year 2005, updated: Tom Meyer is NOT born. 5. Year 2005-2499, updated: No successors of Tom Meyer exists. 6. Year 2500, updated: John Meyer (a successor of Tom Meyer) does not exist . Non-uniform time progression The assumption of a uniform time progression, in Rule1, is a slight simplification to ease the discussion. Support of invariance with respect to coordinate transformation and the possibility of curved space-time (as with GRT) may require more sophisticated assumptions instead. What is really required is that the world at any stage of the evolution has (almost) zero time-like extension. The evolution produces a series of time-slices (i.e., snapshots), but there is always just a single time-slice ...

Context 2

... on Space-time Topology The two rules stated above also have an influence on the type of geometry and topology that may be appropriate for a model of our world. The traditional view of SRT and GRT is that the space-time of our world has (3+1) 3 dimensions, which allows for various types of topological spaces, such as, for example, a 3-sphere, S . Leaving out 1 space dimension, the 2 space dimensions + 1 time dimension may be graphically represented as shown on the left hand side of Fig. 1. The right hand side of Fig. 1 shows the situation when the above two rules for the "Process of World State Progression" are applied. Instead of a topological space that is continuously expanding in the direction of time, we have a series of "time-slices". Rule2 says that there always exists only a single time-slice. The old time-slice is discarded when a new one is generated. If, however, the extension of the world in the time dimension is always equal to 0, one may question whether this should be viewed as a dimension at all; at the least, the time dimension has quite different characteristics than the space dimensions. One possible argument for keeping a time dimension is that it makes sense to label the series of time slices by unique time coordinates. However, if this is the justification for the assumption of a non-zero extension of the time dimension and Rule2 is assumed to hold, it should be made clear that the state of the world of past time slices is empty. When "time dimension" and "time coordinates" are discussed within this paper, it must be clarified that this always refers to the time represented by the time coordinates of the world's space-time, as opposed to the "proper time" known from SRT and GRT. Inclusion of the proper time in these discussions is outside the scope of this paper because it would not change the major conclusions of the paper (Note 1). Although the above described rules for the evolution of the world seem to be in accordance with many implicit assumptions in various areas of physics, some alternatives are discussed in this section . 5.1 Old states are not discarded - the past remains in existence Keeping past world states would seem to be a prerequisite for any time travel function. Let us assume that the past is retained in the form of a series of snapshots similar to a sequence of pictures contained in a movie. The follow-up question becomes this: what is the role and purpose of these world state snapshots with respect to the evolution of the world? Specifically, when and how may these snapshots become activated to play a role in the world's evolution? If the snapshots would never be used, their purpose (and existence) are highly questionable. Assuming that the snapshots are maintained just to enable time travel would be rather silly. Disregarding their oddness, there are two main alternatives recognized by the author for the way in which these potential past snapshots may be activated or used: 1. The snapshots remain permanently active, which means they are a permanent source for interpretations by the physics-interpreter. The continuous generation of new worlds at all stages of history would be the result. This is similar to the new worlds occurring in Everett's "many-worlds interpretation" of quantum theory (see Everett (1957)). In fact, such a scheme might be defined as an extension of Everett's many-worlds interpretation. 2. A modification of a past snapshot (e.g., as a consequence of a time travel function) automatically starts an update of the world by replacing all the snapshots that are younger than the modified snapshot up to the youngest (i.e., the present) state. In other words, changing the past would automatically and instantly result in a replacement of the present. As intended, these two alternatives for keeping the past would enable time travel (forgetting for the moment about some other physical problems). Notice, however, that both alternatives for utilizing the past do not result in the time travel paradox. With the "many-worlds process" (alternative 1), the non-existence of Tom Meyer would occur only in newly generated parallel worlds, which would not be a problem for the "old" worlds. With the "automatic update of the present" (alternative 2), the paradox is avoided because the contradicting statements occurring in the example described in the Introduction are now produced in sequence. Facts that would contradict when assumed concurrently may cause no problem when they occur in a certain sequence. The sequence would now be as follows: 1. Year 2005: Tom Meyer, a son of Tim Meyer, is born. 2. Year 2500: John Meyer, a successor of Tom Meyer, lives. 3. Year 2000, resumed: John Meyer performs actions that prevent the birth of Tom Meyer. 4. Year 2005, updated: Tom Meyer is NOT born. 5. Year 2005-2499, updated: No successors of Tom Meyer exists. 6. Year 2500, updated: John Meyer (a successor of Tom Meyer) does not exist . Non-uniform time progression The assumption of a uniform time progression, in Rule1, is a slight simplification to ease the discussion. Support of invariance with respect to coordinate transformation and the possibility of curved space-time (as with GRT) may require more sophisticated assumptions instead. What is really required is that the world at any stage of the evolution has (almost) zero time-like extension. The evolution produces a series of time-slices (i.e., snapshots), but there is always just a single time-slice in existence. As a result of ...