Deeper Relativity Budget Scans

Time Length Reciprocal Three-Row Scan

one gamma, two ledgers

Pairing time dilation with length contraction shows one gamma multiplying time while dividing length. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One gamma lengthens time and shortens length

For gamma 5/4, proper time 4 s becomes lab time 5 s. The matching rest length 5 m becomes moving length 4 m.

4⋅54=55÷54=44\cdot\frac{5}{4}=5\quad5\div\frac{5}{4}=4
Reciprocal time-length rowThe row carries both checked time dilation and checked length contraction.speedquiet leglimitbetagammaproper timelab timerest lengthmoving length

Doubling both source measurements doubles both outputs

The second row doubles the source numbers: proper time 8 s and rest length 10 m. The checked outputs are 10 s and 8 m.

8⋅54=1010÷54=88\cdot\frac{5}{4}=10\quad10\div\frac{5}{4}=8
Reciprocal time-length rowThe row carries both checked time dilation and checked length contraction.speedquiet leglimitbetagammaproper timelab timerest lengthmoving length

Tripling the first row keeps the reciprocal pattern

The third row scales the same pair again. Proper time 12 s maps to lab time 15 s, while rest length 15 m maps to moving length 12 m. The same gamma appears in both ledgers.

12⋅54=1515÷54=1212\cdot\frac{5}{4}=15\quad15\div\frac{5}{4}=12
Reciprocal time-length rowThe row carries both checked time dilation and checked length contraction.speedquiet leglimitbetagammaproper timelab timerest lengthmoving length