Expanding on the previous post here, this is a summary of the equations of motion for space-time and time-space. See also the Time-space Glossary option above.
r = displacement magnitude, t = time magnitude, v = velocity, v_{0} = initial velocity, a = acceleration, ℓ = celerity, ℓ_{0} = initial celerity, b = deprestination
Space-time (3+1) | Time-space (1+3) | |
Assumption | da / dt = 0 | db / dr = 0 |
da / dr = 0 | db / dt = 0 | |
First equation of motion | dv / dt = a | dℓ / dr = b |
v = v_{0} + at | v = v_{0} + br | |
Second equation of motion | dr / dt = v = 1/ℓ | dt / dr = ℓ = 1/v |
r = r_{0} + v_{0}t + ½at² | t = t_{0} + ℓ_{0}r + ½br² | |
Third equation of motion | dv / dr = a/v = aℓ | dℓ / dt = b/ℓ = bv |
v² = v_{0}² + 2a(r – r_{0}) | ℓ² = ℓ_{0}² + 2b(t – t_{0}) | |
2a = (v² – v_{0}²) / (r – r_{0}) | 2b = (ℓ² – ℓ_{0}²) / (t – t_{0}) | |
Relation | dv / dℓ = a/bv = aℓ/b | dℓ / dv = b/aℓ = bv/a |