Heidegger vs Einstein

Heidegger, in his 1924 lecture on the 'Concept of Time', said 

The explicit question concerning the 'essence' of time ordinarily remains within the everyday experience of time.

The essence of a thing is what is true it in all possible worlds. Our lived experience of it can't be its essence if there is some other way it could be. The only lion I ever met was very friendly and gentle. But the essence of a lion isn't friendly or gentle.  Newtonian 'absolute' Time might be thought off as being the same everywhere. But our lived experience is that sometimes it goes by quickly and sometimes it drags on at a snail's pace. Newton's 'absolute' time proceeds at the same pace with respect to any mechanical or biological entity. But Einstein had shown this was not true of clocks or twins if they were separated and subjected to acceleration. 

Time is the heavens or, rather, their revolving motion; time is movement.*

Newton considered the "fixed stars" an approximation of an absolute, non-accelerating frame of reference (absolute space) against which true motion and inertial behavior are measured. While Newton believed in a fixed absolute space, the distant stars provide a practical, stationary background to determine if a body is experiencing net forces. Sadly this had been experimentally refuted by the time Heidegger wrote this.

It is clear from both statements that one looks for time in those things one refers to when one specifies the 'Then' of one's concerns on a daily basis: the heavens and the course of the sun.

Germans had calendars & clocks & watches by then.  

The first surviving treatise on time, whose findings have subsequently been, and still are, highly authoritative, namely Aristotle's discussion in his Physics (ontology of the world),

No. Physics is from φύσις (physis), meaning "nature," "origin," or "the essential quality of natural things". The word ontology is not Aristotelian. It dates from the 17th century & refers to Metaphysics- that which is beyond Nature. 

also adheres to the most common way of encountering time.**

We don't encounter it any more than we encounter gravity or oxygen.  

Aristotle calls to mind the state of affairs addressed in these statements and concludes: although time is not movement, it is nevertheless part and parcel of what is moved.

He lived long ago. What he called to mind was ignorance from the perspective of the Twentieth Century.  

What is time itself?

It appeared to be the fourth dimension of Space-Time. Might there be yet more 'infolded dimensions'? Kaluza & Klein had begun thinking along these lines by 1921.  

The possibility of highlighting the phenomenon of time and grasping it ontologically in light of this given state of affairs  requires us first to understand 'movement' ontologically.

This is done geometrically.  

Aristotle discovered movement as an ontological characteristic of entities and conceptualized it ontologically.

No.  He defined motion (kinesis) as the actuality of potentiality—the process of a being fulfilling its capacity to change. He viewed all change, including location, growth, and quality alteration, as driven by natural tendencies for objects to seek their rightful, restful place. This is what we would call teleology, not ontology. 

Compared with Plato,

who was right to emphasize the importance of geometry 

he reached a more original ground within the same research project.

No. He was saying 'things move because it is in their nature to do so'. Heavy things will fall faster than light things because the nature of heaviness is such. 

For the first time, this opened up the possibility of delineating 'time' ontologically.

Delineation is geometric. It is the establishment of a metric. Aristotle wasn't mathsy.  

... when Dasein

According to Heidegger- the Dasein is the human place of spatialization and temporalization – the place where space becomes and is space and time becomes and is time

explicitly inquires into the essence of time,

it doesn't do any such thing. It is a useless lump of shit. Time & Space are dimensions in a geometrical description of the Universe. They have a physical interpretation. This changes when new data becomes available. But interpretations aren't the thing itself. My interpretation of the Bible has changed over time. I no longer believe, as Pope John Paul suggested to me, that the central teaching of the Sermon on the Mount is 'fuck the Police!' But the Bible itself hasn't changed. 

it puts forward questions and solutions in terms of presencing concern . But this reckoning with time never turns it into 'space'.

I suppose, if Time were absolute, it would exist even where no Space existed. But Liebniz, not Newton, was on the right track in this matter. Kant sought for a priori reasons why Newton must be right. But he wasn't right. He was wrong.  

Time cannot be spatialized.

Yes it can if the multiverse hypothesis is true and there is a metric based on Lewis-Stalnacker 'closest possible worlds'. This is why Spiderman can visit universes where Dr. Strange is a cat.  

The analysis of the ontological meaning of the clock

or the cat 

and Aristotle's interpretation; have shown that the calculational approach to time is a particular kind of temporalization [Verzeitlichen] in the mode of presencing [ Gegenwdrtigen ].

Nonsense! Calculating the age of a dinosaur fossil doesn't make dinosaurs 'present'. Otherwise they would eat the guy doing the calculation.

Though one allows 'non-reversibility' as a distinct predicate of time,

Classical Physics doesn't. There could be a 'Maxwell demon' reversing entropy.  

one does so on the understanding that one would much rather reverse time's direction, that is, that one would very much like to repeat and retrieve time and have it completely available in the present moment as something present -at-hand.

Nobody wants any such thing. Time-travel, however would be cool.  

Heidegger was aware of Einstein's theory

The current state of this research is established in Einstein's relativity theory. Some of its propositions are as follows: Space is nothing in itself; there is no absolute space.

Leibniz was on the right track. Einstein's general theory gives us equations for the geometry of Space Time. However, things like the 'cosmological constant' had to be changed when empirical evidence suggested the Universe was expanding. 

It exists merely by way of the bodies and energies contained in it. (An old proposition of Aristotle's:)

We don't know why it exists or, rather, we don't know the precise mechanism by which the Big Bang occurs- though there are some very promising theories. Aristotle rejected the 'void' or 'vacuum' which Nature abhors. He also believed matter to be eternal. 

Time too is nothing.

It is a dimension of Space-Time geometry.  

It persists merely as a consequence of the events taking place in it. There is no absolute time, and no absolute simultaneity either.

But there is absolute bullshit. Einstein says he is offering a physical interpretation- not a metaphysical one- of equations which, it turned out, had superior predictive power. They were accepted because it was very useful to have more accurate predictions. 

In seeing the destructive side of this theory, one readily overlooks what is positive about it, namely, that it demonstrates precisely the invariability, with respect to arbitrary transformations, of those equations describing natural processes.

It turned out, that wasn't the case. If there was a Big Bang then there would be quantum effects at the cosmological level. A 'grand unified theory' combining General Relativity & QMT would be required. 

Heidegger wrote

Here we shall not go into the problem of the measurement of time as treated in the theory of relativity.

 There is no such problem. Clocks remain clocks. Twins remain twin. It is just that one clock or twin is sent on a round trip journey under high acceleration, when they return they will show less elapsed time than their counterpart. One clock will be behind the other. One twin will have aged less. 

If the ontological foundations of such measurement are to be clarified,

There is no unique metaphysical foundation for a purely physical interpretation.  

this presupposes that world-time and within-time-ness have already been clarified in terms of Dasein's temporality,

Because shit is clarified to shit by shit.  

and that light has also been cast on the existential-temporal Constitution of the discovery of Nature

That light would involve learning tensor calculus not talking nonsense. 

and the temporal meaning of measurement.

which is its utility. What Einstein was doing enabled Astronomers etc. to make better predictions. That was useful.  

Any axiomatic for the physical technique of measurement

there are heuristics. There is are no axiomatics.  

must rest upon such investigations, and can never, for its own part, tackle the problem of time as such.

Because the problem of shit can only be tackled by talking stupid shit.