Lorentz tranformations and time slices

  • #1
I have a question about the way lorentz transformations work with respect to a 'block' view of the universe.

Take our universe as a 3d chart with 2 space dimensions and one time dimension (ignoring the other space dimension for simplicity). You chart it using some "god's eye view" reference frame, or just any single reference frame. So you end up with simultaneous events perpendicular to the time axis for any point in time. Each time slice is just a rectangle showing that moment in time (for the 'god-like' observer).

Now when you apply lorentz transformations for another observer, can you use that same chart? Is it true that the new observer just has a slightly angled time slice (due to the transformation) but still uses the same chart? So their time slices are stretched rectangles compared to the god-like observer?

Or do lorentz transformations change things so much that it would be impossible to represent the universe using a single 4d block (i.e. you'd need a different 4d block for each observer)?
 

Answers and Replies

  • #2
422
3
Lorentz transformations account for time dilation AND length contraction, so not only is the time axis different according to observers in different inertial reference frames, but the spatial axes also changes. Perhaps this video might be helpful for helping you visualize:


So I think It's more of the latter suggestion, it's not just the time axis that is stretched relative to an outside "god-like" observer, but in fact, the entire space that is distorted.
 
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  • #3
Thanks that is a good video.

I guess my question is about the way a time slice deforms in the 'worst case'. For the 3d space-time example it would appear the time slices get transformed from a regular rectangle into a less regular rectangle. But would the rectangle have any curves in it or would it stay 'flat'?

For example could you always represent an observer's transformation as parabola or can it get more complex than that (for a 1d space, 1d time example)? When I look at the lorentz transformation formula (with a very untrained eye):

78195e8f63116bf11b2bbef574fbcc25.png


It seems simple enough that it wouldn't 'warp' the time slice in a very complex way. I just wonder what the limit of that warping is.
 
  • #4
422
3
You're right, the warping is really not that complex. It's linear, in fact.
 
  • #5
30,143
6,602
It is the same space and the same metric, just different basis vectors for different observers.
 

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