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, v0 = initial velocity, a = acceleration, w = lenticity, w0 = initial lenticity, b = relentation.
| 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 | dw / dr = b |
| v = v0 + at | v = v0 + br | |
| Second equation of motion | dr / dt = v = 1/w | dt / dr = w = 1/v |
| r = r0 + v0t + ½at² | t = t0 + w0r + ½br² | |
| Third equation of motion | dv / dr = a/v = aw | dw / dt = b/w = bv |
| v² = v0² + 2a(r – r0) | w² = w0² + 2b(t – t0) | |
| 2a = (v² – v0²) / (r – r0) | 2b = (w² – w0²) / (t – t0) | |
| Relation | dv / du = a/bv = aw/b | du / dv = b/aw = bv/a |