Physical time is the time dimension we derive empirically from our study of a universe compounded of space, time, matter, gravity, radiation and whatever else happens to be out there, dark energy, dark matter and anti-matter, and so on. The origins of this quantitative time dimension can be traced back to the Big Bang. The independence of the speed of light from the local reference frame means that this kind of time is another spatial dimension in a 4-dimension space-time geometry. This identification is not changed, only modified, when gravitation generalizes the pseudo-Euclidean geometry of a Minkowski space to a 4-dimensional Riemannian manifold of inertial signature (- + + +)
Phenomenologically, the defining distinction between space and time is that transformations from one location to another, in all 3 space directions are reversible, while temporal location is not. (One must exercise some caution in the way this is stated, because to speak of "motion" in time does violence to logic and language: time is the measure of motion (Aristotle) and does not itself move.)
The Noether Theorems state that every symmetry in the laws of nature leads to a conservation principle: reversibility of spatial location leads to the conservation of momentum, while time reversibility leads to the conservation of energy. However, if one looks at daily experience, and not just equations in which a variable t can be replaced by a variable -t, time is not reversible. Yet the conservation of energy (generalized in relativity to the conservation of mass-energy) has been verified by empirical observation. (For a delightful demonstration of this, take a look at the energy experiment devised in Phillip Morrison's WGBH TV series, The Ring of Truth,("Change"))
This suggests that, at one point, physical time was exactly like all the other spatial dimensions. ( Paradoxically this does not mean that one could "move" in time, but that either (i) there was no motion, or (ii)there exists another "intrinsic" 5th dimension which assumed the role that time has in our universe, relative to which motions in all 4 dimensions were equally possible). This suggests that we entertain the hypothesis that physical time is a Broken Symmetry on one of the directions in this primordial 4-dimensional Space, and that the breaking of this symmetry is what we have observed as the evidence for a Big Bang 14 billion years ago. The symmetry in question is of course time reversal: it is why one can replace t by -t in the laws of mechanics but not in real life.
The geometry of this primordial 4-dimensional space would have been governed by the quaternion group, with vectorial directions v = ct +ix +jy+kz. Note that any two-dimensional linear subspace of such a geometry is a 2D Minkowski space. What we call going forward in time is really time's inability to go backwards!
How does physical time relate to experiential or psychological time? Thinking along these lines leads to surprising conclusions, and may give even so wrong-headed an opinion as that of Peter Lynds a smidgen of respectability. (See Peter Lynds
It is an error of logic to speak of the physical time that we observe in our universe as "flowing", which is what people like Peter Lynds maintain. Events flow through time. In the same way, one does not say that space flows through space, but that events or entities are located in space. Psychological time however, is essentially a cyclic process, the arising and passing away of consciousness, the pulse of the mechanisms of birth, death and rebirth that is inherent to life itself and to all living or sentient beings.
The emergence of consciousness, its passing away, and subsequent re-emergence is the most elementary of all the observations of psychology. It is inevitable that one would be inclined extrapolate from this to material transmigration, or re-incarnation, (though not in the same body). We do not want to enter into this debate here.
What we observe or call the "flow" of time arises from the combination, fusion, interaction, call it what one will, of these two distinct phenomena, both of which are subsumed under the same word, time. When the time dimension of consciousness is conjoined with the time dimension of the physical universe, a cyclic process is imposed upon an irreversible spatial dimension, thus giving rise to the appearance of a temporal flow through past, present and future.
It is therefore not a violation of logic and language to say that the "flow of time" is an observed phenomenon, provided that such statements include the observer him/herself as a subject of the fundamental psychic process of arising, passing away, and re-emergence of consciousness.