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MetricFirst.png (485 × 502 pixels, file size: 25 KB, MIME type: image/png)

Summary

Description
English: The metric-equation's single map-frame (top) can facilitate many useful quantitative insights before invoking the Lorentz-transform's pair of frames[1], each with their own yardsticks and synchronized-clocks (bottom). The time-dilation and length-contraction effects depicted for the green moving map-frame at bottom, which use simultaneity defined by the more colorful stationary map-frame, correspond to a proper-velocity dx/dτ of 1 [ly/ty], a Lorentz factor dt/dτ of Sqrt[2], or a coordinate-velocity dx/dt of 1/Sqrt[2] [ly/y].
Date
Source Own work
Author P. Fraundorf

Further discussion

This figure was inspired by a collection of such clock-lattice figures in the literature and on wikipedia.

Code

This is Mathematica code used to generate these figures.

The top half was generated by: 

chw = .1; cra = .2; rhw = .05; tangle = 
Pi/6; offset = {1.414, .5, .5}; metricmap = Graphics3D[
 {
  Arrowheads[Medium],
  Cylinder[{{0, -chw, 0}, {0, chw, 0}}, cra], 
  Cylinder[{{0, -chw, 0} + {0, 0, 1}, {0, chw, 0} + {0, 0, 1}}, cra],
  Cylinder[{{0, -chw, 0} + {0, 1, 0}, {0, chw, 0} + {0, 1, 0}}, cra],
  Cylinder[{{0, -chw, 0} + {1, 0, 0}, {0, chw, 0} + {1, 0, 0}}, cra],
  (* Cylinder[{{0,-chw,0}+{2,0,0},{0,chw,0}+{2,0,0}},cra], *)
  Tube[{{0, 0, 0}, {0, 0, 1}}, rhw],
  Tube[{{0, 0, 0}, {0, 1, 0}}, rhw],
  Tube[{{0, 0, 0}, {1, 0, 0}}, rhw],
  Arrow[{{0, -chw, 0}, {0, -chw, cra}}],
  Arrow[{{0, -chw, 0}, {cra Sin[tangle], -chw, cra Cos[tangle]}}],
  Arrow[{{0, -chw, 0} + {0, 0, 1}, {0, -chw, cra} + {0, 0, 1}}],
  Arrow[{{0, -chw, 0} + {0, 0, 1}, {cra Sin[tangle], -chw, 
      cra Cos[tangle]} + {0, 0, 1}}],
  Arrow[{{0, -chw, 0} + {0, 1, 0}, {0, -chw, cra} + {0, 1, 0}}],
  Arrow[{{0, -chw, 0} + {0, 1, 0}, {cra Sin[tangle], -chw, 
      cra Cos[tangle]} + {0, 1, 0}}],
  Arrow[{{0, -chw, 0} + {1, 0, 0}, {0, -chw, cra} + {1, 0, 0}}],
  Arrow[{{0, -chw, 0} + {1, 0, 0}, {cra Sin[tangle], -chw, 
      cra Cos[tangle]} + {1, 0, 0}}],
  Lighter[Green],
  Cylinder[{{0, -chw, 0} + offset, {0, chw, 0} + offset}, cra],
  Black,
  Arrow[{{0, -chw, 0} + offset, {0, -chw, cra} + offset}],
  Arrow[{{0, -chw, 0} + 
     offset, {cra Sin[tangle], -chw, cra Cos[tangle]} + offset}],
  Green,
  Arrowheads[Large],
  Arrow[Tube[{offset, offset + {1, 0, 0}}, .01]],
  Black,
  Text[Style[
    "(c\[Delta]\[Tau]\!\(\*SuperscriptBox[\()\), \
\(2\)]\)=(c\[Delta]t\!\(\*SuperscriptBox[\()\), \(2\)]\)-\[Delta]\!\(\
\*OverscriptBox[\(x\), \
\(\[RightVector]\)]\)\[CenterDot]\[Delta]\!\(\*OverscriptBox[\(x\), \
\(\[RightVector]\)]\)", FontSize -> 16], offset + {0, 0, .4}]
  }, Boxed -> False, (* PlotLabel->"Map Frame and Traveler Clock",*)
 BaseStyle -> {FontFamily -> "Courier", FontSize -> 12}, 
 ImageSize -> 512
 ]

The bottom half was generated by: 

chw = .1; cra = .2; rhw = .05; tangle = 
Pi/6; offset = {1.414, .5, .5}; mapframes = Graphics3D[
 {
  Arrowheads[Medium],
  Cylinder[{{0, -chw, 0}, {0, chw, 0}}, cra], 
  Cylinder[{{0, -chw, 0} + {0, 0, 1}, {0, chw, 0} + {0, 0, 1}}, cra],
  Cylinder[{{0, -chw, 0} + {0, 1, 0}, {0, chw, 0} + {0, 1, 0}}, cra],
  Cylinder[{{0, -chw, 0} + {1, 0, 0}, {0, chw, 0} + {1, 0, 0}}, cra],
  (* Cylinder[{{0,-chw,0}+{2,0,0},{0,chw,0}+{2,0,0}},cra], *)
  Tube[{{0, 0, 0}, {0, 0, 1}}, rhw],
  Tube[{{0, 0, 0}, {0, 1, 0}}, rhw],
  Tube[{{0, 0, 0}, {1, 0, 0}}, rhw],
  Arrow[{{0, -chw, 0}, {0, -chw, cra}}],
  Arrow[{{0, -chw, 0}, {cra Sin[tangle], -chw, cra Cos[tangle]}}],
  Arrow[{{0, -chw, 0} + {0, 0, 1}, {0, -chw, cra} + {0, 0, 1}}],
  Arrow[{{0, -chw, 0} + {0, 0, 1}, {cra Sin[tangle], -chw, 
      cra Cos[tangle]} + {0, 0, 1}}],
  Arrow[{{0, -chw, 0} + {0, 1, 0}, {0, -chw, cra} + {0, 1, 0}}],
  Arrow[{{0, -chw, 0} + {0, 1, 0}, {cra Sin[tangle], -chw, 
      cra Cos[tangle]} + {0, 1, 0}}],
  Arrow[{{0, -chw, 0} + {1, 0, 0}, {0, -chw, cra} + {1, 0, 0}}],
  Arrow[{{0, -chw, 0} + {1, 0, 0}, {cra Sin[tangle], -chw, 
      cra Cos[tangle]} + {1, 0, 0}}],
  Lighter[Green],
  Cylinder[{{0, -chw, 0} + offset, {0, chw, 0} + offset}, cra], 
  Cylinder[{{0, -chw, 0} + {0, 0, 1} + 
     offset, {0, chw, 0} + {0, 0, 1} + offset}, cra],
  Cylinder[{{0, -chw, 0} + {0, 1, 0} + 
     offset, {0, chw, 0} + {0, 1, 0} + offset}, cra],
  Cylinder[{{0, -chw, 0} + {1, 0, 0} + 
     offset, {0, chw, 0} + {1, 0, 0} + offset}, cra],
  Tube[{{0, 0, 0} + offset, {0, 0, 1} + offset}, rhw],
  Tube[{{0, 0, 0} + offset, {0, 1, 0} + offset}, rhw],
  Tube[{{0, 0, 0} + offset, {1, 0, 0} + offset}, rhw],
  Black,
  Arrow[{{0, -chw, 0} + offset, {0, -chw, cra} + offset}],
  Arrow[{{0, -chw, 0} + 
     offset, {cra Sin[tangle], -chw, cra Cos[tangle]} + offset}],
  Arrow[{{0, -chw, 0} + {0, 0, 1} + 
     offset, {0, -chw, cra} + {0, 0, 1} + offset}],
  Arrow[{{0, -chw, 0} + {0, 0, 1} + 
     offset, {cra Sin[tangle], -chw, cra Cos[tangle]} + {0, 0, 1} + 
     offset}],
  Arrow[{{0, -chw, 0} + {0, 1, 0} + 
     offset, {0, -chw, cra} + {0, 1, 0} + offset}],
  Arrow[{{0, -chw, 0} + {0, 1, 0} + 
     offset, {cra Sin[tangle], -chw, cra Cos[tangle]} + {0, 1, 0} + 
     offset}],
  Arrow[{{0, -chw, 0} + {1, 0, 0} + 
     offset, {0, -chw, cra} + {1, 0, 0} + offset}],
  Arrow[{{0, -chw, 0} + {1, 0, 0} + 
     offset, {cra Sin[tangle], -chw, cra Cos[tangle]} + {1, 0, 0} + 
     offset}],
  Green,
  Arrowheads[Large],
  Arrow[Tube[{{.5, .5, .5} + offset, {.5, .5, .5} + 
      offset + {1, 0, 0}}, .01]],
  Black,
  Text[Style[
    "(x',y',z',t')=(\[Gamma](x-vt),y,z,\[Gamma](t-\!\(\*FractionBox[\(\
vx\), SuperscriptBox[\(c\), \(2\)]]\)))", FontSize -> 16], 
   offset + {0, 0, 1.3}]
  }, Boxed -> False, (* PlotLabel->"Map Frame and Traveler Clock",*)
 BaseStyle -> {FontFamily -> "Courier", FontSize -> 12}, 
 ImageSize -> 512]

Footnotes

  1. ↑ P. Fraundorf (2011/2012) "Metric-first & entropy-first surprises", arXiv:1106.4698 [physics.gen-ph].

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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