A loop can describe ordinary feedback, a simultaneous consistency condition, or a proposed journey back to an earlier event. These are different mathematical and physical questions, even when their diagrams contain similar arrows.
Feedback progresses through time
A thermostat measures a temperature, changes a heater setting and later measures the resulting temperature. The variables influence one another, but each response occurs after the input that produces it. Retaining time labels turns the apparent two-variable loop into a sequence of different events.
For an original dimensionless illustration, let x at the next tick equal 0.5x + 1. From x = 0 the sequence is 0, 1, 1.5, 1.75 and so on. It tends to the fixed point x = 2. The recurrence produces a succession of states; the equilibrium equation x = 0.5x + 1 identifies a state that would remain unchanged. Solving that equation does not describe the time required to approach it.
Change the multiplier to −0.5 and the approach oscillates around 2/3. Change it to 1.2 and the fixed point is −5, but nearby departures grow rather than decay. Existence of a self-consistent state and dynamical stability are separate properties.
A closed condition can have zero, one or many solutions
Suppose a binary value must return to itself through a rule f. Consistency requires x = f(x). The identity rule has two solutions, zero and one. A constant-zero rule has exactly one. A bit-flip rule has none. No physical interpretation is needed to calculate these cases.
If no solution exists, the complete set of stated conditions cannot all hold. If several solutions exist, consistency alone has not selected one. If exactly one exists, a further question remains: what preparation and dynamics could realize it? These three outcomes should not be conflated with the possibility, impossibility or likelihood of an actual time machine.
Allowing probability distributions changes the problem. A fair distribution is invariant under a bit flip even though no definite bit is fixed by that operation. This solves a distributional consistency equation; it does not create a definite classical history satisfying the original bit constraint. A theory must state which objects its consistency rule applies to.
A spacetime loop is a stronger proposal
A closed timelike curve is a path that can be followed locally in a timelike direction while returning to its starting event in the spacetime description. Its analysis concerns spacetime geometry, allowed matter and global initial-value structure. An ordinary controller with delayed feedback does not provide such a curve.
Hawking's chronology-protection work investigates physical obstacles to creating closed timelike curves under specified circumstances. It motivates restrictions on the admissible geometry and matter; its conjecture should not be promoted into a theorem covering every conceivable spacetime and quantum theory. Conversely, writing a geometrical example does not establish that it can be formed from realistic preparations.
Care is also needed with time-symmetric equations. Reversing a mathematical solution or imposing a future boundary condition is not by itself an operational procedure for changing an already recorded past. One must specify which boundary data can be independently controlled and whether the full problem remains consistent.
Which changes are admissible?
In an ordinary acyclic causal model, replacing one local rule can define an intervention while leaving the rest intact. In a cyclic model, the replacement may alter the number of global solutions. A proposed experiment therefore needs a rule for solving the modified system and, where necessary, selecting among its solutions.
For example, two constraints x = y and y = x admit two binary assignments. Adding x = 0 selects one. Replacing the second relation by y = 1 − x makes the pair inconsistent. The algebra identifies the conflict; it does not tell us which physical component yields, fails or changes its governing equation in an attempted implementation.
Keep internal recurrence distinct from backward signalling
SMM/DTT may describe recurrent inner operations, completion cycles or histories subject to global restrictions. Such structures should be defined on their own terms. A recurrence in inner generation is not evidence of a controllable message arriving earlier in outer physical time.
The expanded book's order and physical clocks and completion bound offer useful checks: identify the observable timing, specify permitted transfers and calculate the consequences of changing a preparation. This allows consistency questions to sharpen the programme without treating a suggestive loop diagram as a physical result.
Sources and further reading
- S. W. Hawking, Chronology protection conjecture, Physical Review D 46 (1992), 603–611.
- Must a cause precede its effect? The recurrence and binary consistency examples here are original teaching calculations.