Timeline for Questions about time and space (from beginner)
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 25, 2014 at 12:36 | comment | added | Stan Liou | I don't disagree that they're different concepts; I rather, I'm saying that those mappings trivially match the concept of projection, especially in STR where we in an inertial frame we are free to treat each slice as the same space, but also in the more general case because the concept doesn't have such restrictions. Though I suppose I see why you wouldn't want to bring it up. | |
| Apr 25, 2014 at 12:08 | comment | added | Gerald | @StanLiou One more approach: A space can be described as a disjoint union of slices, or as a product of projections; it's conceptually different. | |
| Apr 25, 2014 at 11:54 | comment | added | Gerald | @StanLiou ...Slices frequenty can be embedded injectively into the space, whereas projections usually aren't injective. | |
| Apr 25, 2014 at 11:50 | comment | added | Gerald | @StanLiou Using the term 'projection' to decompose e.g. a vector space into its components, or a bit more general decomposing a cartesian product into its components, including decomposing the points/elements of the space/set into coordinates is ok. That way projections span the product space. But applying the projection map to the whole 4D space, including contents, would map different objects of the space to the same point in the codomain, in contrast to a slice. Take a loaf of bread as an example: Mapping all points of the loaf to a 2D-plane is different from a slice of bread. | |
| Apr 24, 2014 at 23:14 | comment | added | Stan Liou | I suppose I'm confused about your unwillingness to use the term "projection" because you seem to be using it in a sense I don't recognize. I don't disagree with what you just said, but in what you describe, the mapping from spacetime to either the time or the spatial submanifold would both be projections. | |
| Apr 24, 2014 at 21:08 | comment | added | Gerald | @StanLiou I agree thus far. But we've still to slice the 4D-spacetime to get the observed spatial 3D-manifold (including contents) for a fixed time in the respective inertial frame. | |
| Apr 24, 2014 at 17:56 | vote | accept | Michal | ||
| Apr 24, 2014 at 15:46 | comment | added | Stan Liou | You're correct in that the Lorentz transformation would be a change of basis, but for orthonormal bases appropriate to an inertial frame, the change of basis matrix itself consists of projection components $\langle e'_i,e_j\rangle$. Moreover, anytime one expresses a vector in an orthonormal basis, one take a sum of projections: $u = \sum_k e_k\langle e_k,u\rangle$, so different inertial frames disagreeing on the time component of a vector is exactly the statement that the projections to their respective temporal axes are different, and so forth for length contraction. | |
| Apr 24, 2014 at 13:18 | comment | added | Gerald | @StanLiou I'd call this a 'change of base'. A projection would be kind of integrating along one or more dimensions, and get kind of lower-dimensional shadow. But no matter, whether we 'see' or 'observe' the universe at one instant, we consider just a lower-dimensional sheet or 'slice' of the 4D-space. That's why I feel uncomfortable with the use of 'projection' in this case. Of course, one could project a slice onto a different manifold of the same dimension, in order to get a description of the slice which may be more suitable for some purpose. | |
| Apr 24, 2014 at 4:52 | comment | added | Stan Liou | 'Projection' is a good term here to start with. In STR, time dilation and Lorentz contraction across different inertial frames can be described as due to projections to different temporal and spatial coordinate axes. The real difficulty is that this describes what we observe, which in STR is a very different concept from what we see or otherwise experience through our senses. | |
| Apr 21, 2014 at 11:57 | history | edited | Gerald | CC BY-SA 3.0 |
Added link to Minkowski space, and discussed Hilbert space vs. 5D space
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| Apr 20, 2014 at 21:17 | history | answered | Gerald | CC BY-SA 3.0 |