Assume a planet that always presents one side to the sun. No moons. The orbit of the planet around the star is essentially spherical, very minor and undetectable aberrations. The orbital period of the planet around the star is in multiples of the lifetimes of the sentient beings. That is, no sentient being lives long enough to go through one complete rotation.
NOTE: this question is not about HOW this would happen, it is to be taken as a given that it DID happen.
A sentient species evolves on this planet.
There are no day/night cycles, no seasons, and no cyclical changes in the sky - it would be constantly different throughout one's lifespan. No repeats. Life evolved without any natural circadian rhythm. Plants grew and died independently of any cycle. No defined growing season. A plant could sprout at any time, and die at any time. No menstrual cycles, no estrus cycles in animals. There would be no biological basis on which to establish any. Life would have evolved without any concept of cyclical time passage. As far as life was concerned, everything was eternally homogenous, time wise. Nothing happened in unison.
To these sentient beings, the passage of time is irrelevant. They have no way to compare the passage of time between people. Heart beats are different, respiration is different, one's pace in walking is different. The interval to travel from one point to another is different. It is human experience that what seems like a long period of time to one person is a short period of time for another person. The beings get to where they are going when they get there, without reference to anyone else's travels. They eat when they need to, irrespective of how long it has taken between meals (think of a snake, that can go for very long periods between meals, depending on how big the last meal was). Gestations are never the same length, so even if two beings got pregnant at the same time, the deliveries would never likely be at the same time. Two objects released from a height at the same moment would be observed to arrive at the ground at the same moment, but there would be no indication that if dropped at some other moment they would arrive at the ground in the same time as these two, without some form of quantized time.
Many of our 'Laws of Physics' require time to be quantized. Humans, of course, had to quantize time before we could develop any theories based on the quantization of time. This is a straightforward 'chicken or egg' thing.
It seems to me that if Galileo had no preconceived idea of the measure of the passage of time (through the passage of seasons, day/night cycles, etc) he would not have been able to discover that the periodicity of a pendulum was constant. Depending on his state of mind, sometimes the swing would appear to take forever, sometimes it would appear to be shorter.
It also seems to me that Newton relied on a distinct concept of the passage of quantized time before hand, in order to develop many of his Laws. (The action/reaction thing, and gravitational attraction thing, would be valid, just not quantized as to the passage of time).
Building structural integrity and engineering calculations for stresses have no time component to them, if built by 'Rule of Thumb' (We built the last one this thick, and it is still standing. The other one was built thinner, and it collapsed.)
If this sentient species had no experience that the passage of time was measurable and periodic, would they ever develop a method to quantize it? Would they ever TRY to quantize it, if there was nothing in their environment that was consistently and repeatedly cyclic, that they needed to or even that they could consistently measure? What would that quantization look like?
I submit that this question is NOT the same as this, as many of the answers pertain to some form of naturally occurring cycle observable by humans, and /or are based on human circadian rhythms, which are not experienced by this species. This life form evolved without any ability to determine periodic time spans. Also, that question asks what would they use to quantize time, this asks what would their method of quantizing time look like? Would they even understand that time could be quantized into absolute divisions, that were equivalent between people and between events?
A further corollary question will be something along the lines of 'What would the Laws of Physics look like without time being quantized in the same constant periodic way that humans have quantized it?' But that is NOT in the scope of this question.
The term 'quantized' comes from the digital field, not from quantum mechanics. The first use of quantum, in this regard, was in 1597. For a non-quantum-mechanics elaboration of the term, See Quantization (signal processing)
Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding.
It is impossible for this species to develop any kind of calendar or any kind of measurement for a periodic recurring cycle. No days, months, years, seasons, hunger cycles, menstrual cycles, crop cycles, planting seasons, harvesting seasons, sleep cycles, or any other natural seasonal phenomena'. It takes three or more lifespans for the sky to repeat. The entire life of the planet has evolved without periodic cycles or rhythms. There is nothing to measure it with. There is no way to tell that this current crop is growing faster, slower, or just the same as the last crop. No way to judge or compare the pregnancy duration of the first child, with the length of the pregnancy for the last child. There is no metric, nothing to count, measure, or quantify. Any answer based on comparing the length of a current event to past or future events in any manner that does not end up with a number, or quantity, of some absolutely replicable unit of measurement, of the actual duration of the event, (so for instance two pregnancies, one then and one now, can be compared in length), is immediately discarded. Any comparison to any time duration in the past has to involve some kind of numerical count for the duration of each event.
Way back as far as 1597, the term 'quantum' was used to mean an amount of something. You could have a quantum of wheat, a quantum of distance, a quantum of liquid, a quantum of patience, a quantum of intelligence. a quantum of solace, a quantum of time. Some of these we consider to be measurable, some not. For instance, we talk about someone having more patience than someone else, or that one is 'losing patience', but what is the measure of patience that reduces to zero? For the things that we consider measurable, we have developed a system whereby they can be measured, counted, or quantified. But unless we have the CONCEPT that we can measure it, there has been no real consistent attempt to develop units of measure. The word 'quantum' is sometimes applied to the 'unit' of measurement - a quantum of distance is the foot, or the inch, or the meter, a quantum of weight is the gram, or the ounce, or the stone. Note that each one is different, even for measuring the same thing, so generically the term applied was 'quantum' meaning 'measurable amount'. But not all things that you could have a quantum of, could you have a 'measurable amount' of. Like patience. We can still talk about a quantum pf patience, without really any attempt at measuring, counting, or quantifying it.
Before we can measure, count, or quantify something, we first have to form the concept that it CAN be measured, counted, or quantified. My question is about developing this concept for time. (It could also be asked about temperature as well - we know something can be hotter or colder than something else, butt it took us a long time to develop the concept that the comparison could be MEASURED.) We can understand that time flows, that there are time passages, that the future is not the past, and we are in the present. Having a concept that time passes is not the same as understanding that it is measurable. Can we have the CONCEPT that time is measurable? We can not see, taste, feel, touch, or senese it except through memory, of knowing that the past happened. That is the process I call quantization and Miriam_Webster agrees with me.
Definition of quantize transitive verb
1 : to subdivide (something, such as energy) into small but measurable increments
Okay, so someone appropriated the term 'quantum' and proceeded to minimize it to absurdity, and it is now narrowly defined by some physics purists as ONLY meaning something to do with quantum mechanics. You do not have to agree with the definition or my terminology, you can come up with your own, but it what I am after is NOT the process of measuring, counting, or quantifying something, it is the process of realizing something CAN be measured or quantified in the first place. I do not want to know HOW it is measured, conted, or quantified, just that this species has the understanding and the concept that it CAN be measured, counted, or quantified.
We humans got the concept that time is something that could be measured, from the obvious day/night cycle. Historically, that is the first form of measuring time. We could COUNT days, and COMPARE the count between one event and another. We could RECORD the count, to compare durations in the present to durations in the past. It was not hard to conceive of the periodicity and the regularity of the day/night cycle. We could count the number of days from winter solace to the next winter solace, and see that the count was always the same. We liver f=through enough winter solaces to get the idea. It was natural for us to develop the concept that time could be measured, counted, and quantified.
But what if there was no calendar? Nothing to count? What natural phenomena would develop the concept that time was measurable in any meaningful fashion? ANswering this is fundamental to my corollary question - what would be the limitations on this species on their concepts in physics, based on their concept of the measurability of time? How would they quantify speed, for instance? They could easily compare two people in a race, and determine one was faster than the other, but what would their equations look like, without time? Distance, it is obvious to measure. But time?And a unit for time? Just as improbable, perhaps, for this species as a unit for patience is to ours.
So any answer HAS to include the reason why this species could even conceive that time could be measured in the first place. Again, any answer based on the premise that time HAD to be measured, in order to do something, is immediately disqualified. If there was no concept that time could be measured, there would be no concept that in any particular situation it was needed to be measured. You just don't do any activity that is time dependent if you do not understand the concept of measuring time. 'Meeting someone at such-and-such time' is just something this species would ever think about, because it just couldn't be done. ANything that puts a demand on this culture for time dependency as we know it is just anthropomorphizing Western culture on another alien species. No they DON'T have to behave the same way we do.
Sure, there are a lot of situations in which measuring patience is useful, but in all that, we still have no concept that patience can be measured in any absolute way, so we do not consider it a necessity to measure it absolutely. Would time be the same for this species? If not, what would trigger in them the concept that time could be measurable in a quantized (repeatable, replicable, consistent, understandable, quantifiable, countable, measurable, observable, any term you wish to use) way?
So okay, being specific about the question, and explaining it thoroughly and understandably takes a long time. But it has become obvious that TL:DR methods just don't work for many people.
This did not make it into the previous edit.
Take, for instance, the development of the concept that length could be measured. I can measure a length of cloth with my arm, and mark it. Tomorrow, I can measure it, and find I get the same mark. I can measure ten arm's lengths, cut it off, and put it aside. I can take another piece of cloth, measure off ten lengths, and cut it. I can then compare the two lengths of cloth, and see they are both the same length. I can cut another piece of cloth at twenty arm's length, and notice it is as long as the other two pieces of ten arm's length combined. So I know length can be reliably, provably, replicably be measured. I can validate my measurement against something else. I can compare one measured piece with another that I made before and see that they are the same. I have no problem with the concept that length can be measurable. So absent any form of calendar or recurring periodic cycle, how do I get the same concept that time is measurable, can be quantized or sliced up or marked off so it can counted or quantified (whatever you want to call it) and assigned a numerical value, and this value will be the same for any other period of time of the same duration? Length I can see and touch and feel and store and compare between the past and the future. I can not do that with time.