Time is local, but it is wider and deeper than that alone. With the work I have been doing on the nature of time from the perspective of theoretical sociophysics time can best be understood as the emergent from the interaction of spacetime, meantime and placetime. Within these three elements of time, 'meantime' is the local, with placetime the regional time around us, human recorded and potential time with spacetime the deepest layer providing the ultimate potential for causality to take place.
So reference to time being local is a valid although incomplete claim. If we take a clock, which most people can read and we can assert they have a 'theory' of how it works. However, I would guess most people (me included) don't know how the clock's inner workings work. This is the difference between the face value and the workings value. If we turn time on it's side (just like a clock or watch) we can see that there is an underlying structure with layers of interconnected parts generating the complexity that we see in the relatively smooth operation at face value level. This kind of sounds like the Newtonian view but this system can quicken over time, to a degree.
All my work points towards a Universe of three phases where the initial stage of physicality is an unfolding temporal sequence, but this is not time. Over the course of that sequence the expanding number of elements and cosmic interactions settle down into systems of patterned structure that physics and cosmologists now study as 'timing' deepens in the Universe. This is still not time. As a consequence of this timing and enough temporal sequence the conditions emerge for the emergence and evolution of life, yet time has still not started although we can see the beginning of primordial time and then proto-time.
Time (proper) doesn't fully begin till behaviourally modern humans emerge and extend from the evolutionary process. Human expression is a perpetual, and very particular causal extension which creates time as a new way of engaging with the physical and terrestrial world around us: the beginning of meaning. Meaning and time are interwoven, with time 'spin up' and meaning 'spin down' and the more meaning(s) humans create and design the faster time spins and the quicker the world around us seems to pass by. However, to say that time is local and time is only local is to say that there is only meantime, and all my work points to an interaction of spacetime, meantime and placetime from which we have 'time' which we perceive from/through meantime.
The single word that brings this idea of time and meaning as interwoven is 'term' which is:
1. the naming of something, which humans cannot avoid doing, and
2. a period of time.
All human expressions have a time limit on them which means that human experience and the 'being' of human is always one that while real is in flux: realativity. This can be tricky to explain and take in but such is the weight of meaning in/through human expression, and its external effects we can take this understanding of time and meaning being interwoven down to the quantum level and begin to understand time as 'temporal capacity'. As meantime and placetime put more demands on spacetime we can begin to really understand a stretching of 'time' to the point where this elasticity can be taken to a point beyond which the 'timing' underlying the physical and terrestrial worlds is threatened.
Such is the effect of the human artificial world on the terrestrial/natural world I think we are at this stage already and in David Lindenmayer's environmental cautionary book 'On Borrowed Time' the idea of borrowing temporal capacity from the very fabric of reality to 'pay' for the present, the local of meantime is something that can't persist and there will come a point (if it hasn't already happened) that the physical world won't pay out any more and time truly will be stretched to breaking point. However, this could never happen at this deepest of levels so we'd see some of the effects we can witness across a number of indicators in the natural world as a consequence of this tension and breakdown in timing.
With different kinds of 'partial time' like meantime, placetime and spacetime, and with a plethora of disciplines as well as face value (2d) understandings and workings level comprehension (3D and 4D) 'time' will always have a number of perspectives and a number of explanations. However, as we strip away the interconnectedness that enables (and shields) time there is something which is understandable and a level that effects everyone and deepening and popularising this awareness can't come a moment too soon.