We describe time as flowing because experiences succeed one another and clocks keep changing. Turning that description into a physical claim requires identifying what changes relative to what. A comparison of two clocks makes the question precise without dismissing the experience that motivated it.
What would a rate of temporal flow measure?
A river's flow can be measured in metres per second or cubic metres per second. Its rate compares a changing quantity with a clock. Asking how quickly time itself flows requires a second parameter against which the proposed rate is defined.
The expression dt/dt = 1 is a mathematical identity. It supplies no experiment that distinguishes a fast universe from a slow one. Replacing the denominator by another clock reading, dτ, produces a meaningful comparison, but now the question concerns the relationship between two specified clocks.
This distinction also applies to the familiar phrase “time slowed down.” A report about felt duration, a measured clock-rate difference and a change of coordinate labels describe different situations. Each can be investigated, provided its observable is identified.
Construct a comparison with a measurable drift
Let a reference clock read t, and suppose another instrument reports C(t) = t + εt²/(2T). Here T is a fixed duration and ε is dimensionless. This is a deliberately simple drift model, not a proposed universal clock law.
Its relative rate is dC/dt = 1 + εt/T. At t = T, the reading difference is εT/2, while the instantaneous rate difference is ε. An accumulated offset and a rate error therefore need not have the same numerical interpretation.
Choose T = 100 seconds and ε = 0.02. At the end, the drifting clock reads 101 seconds, one second ahead of the reference. Its final rate is 1.02 reference seconds per second. Over the first half of the run it gains 0.25 seconds; over the second half it gains another 0.75 seconds.
Those interval comparisons provide a test. They could reveal instrumental drift, a response to an environmental variable, or a failure of the assumed clock model. Calling the result a change in the flow of time adds little until the mechanism is specified.
Separate clock behaviour from relabelling
Suppose instead that every recorded event is relabelled using s = 2t. A process with position x(t) = vt becomes x(s) = vs/2. Its coordinate speed is halved, but no measured meeting, pulse count or distance comparison has changed.
Consistent relabelling transforms the equations and the readout rules together. If one transforms only the time variable and leaves all coefficients unchanged, one generally defines a different model. This is why mathematical rates must be connected to physical standards.
For a monotone relabelling s = f(t), a trajectory obeying dx/dt = F(x) satisfies dx/ds = F(x)/f′(t). The new expression may look explicitly time dependent even when the original equation was autonomous. Appearance of a variable in an equation is not by itself evidence that the apparatus has changed.
Experience poses an additional question
Memory, anticipation and attention organize experience asymmetrically. We remember records, encounter events and form expectations. An account of that organization needs evidence about experience and its physical conditions; a coordinate transformation alone does not explain it.
Conversely, the fact that a duration feels long does not establish that an atomic clock beside the observer accumulated extra proper time. A study can compare verbal estimates, task performance and clock readings without treating any one of them as the definition of all the others.
It is useful to retain the ordinary language while distinguishing its questions. “Does time flow?” can ask about physical evolution, the availability of new records, or the felt passage of experience. A successful answer should say which of these it addresses.
Give inner and outer time distinct jobs
The book's duality-of-time postulate distinguishes inner generation from the succession of completed records. Its clock discussion explains why an ordering parameter is not automatically a duration in seconds.
For SMM/DTT, this makes the next step constructive: specify a completion rule, a physical clock and the relationship between their parameters. Then calculate a comparison such as the two-clock drift above. A new observable relation would give the temporal proposal empirical content; a relabelling would clarify its description.
Sources and further reading
- The BIPM's definition of the second anchors physical duration to a specified atomic transition frequency. It does not define a metaphysical flow.
- Continue with duration and order and coordinates and observers. The drift calculation is an original teaching example.