What is "time"

Even the multivalued (at edges) square wave is not perfectly reproduced. There is always a small "over shoot." This is Gibb's Effect ( or phenomena ) seen below:

That's because it's a finite approximation of an infinite series.

By "step" I was referring to a discontinuity.
I took it you meant the step function (integral of the impulse).

- Sort of obvious as I said the Fourier transform would have a dot at the mid point,
I thought I understood but maybe not. I assumed you meant "the amplitude of a sample" commonly drawn as a dot.

but I could had said: "The FT of a function which is no where multivalued
"Step function" is clear enough.

(as is the square wave at it edges)
If you say "square wave" esp in connection with the FT it's understood to be the idealization.

and discontinuous is not perfect.
Not sure what you mean here but it's understood the discontinuity in a step function is an idealization.

That "dot" has a name but I forget it so may not be able to search to show you the transform of this function which will have two dots. One at (0,1) and the other at (1,1). Here is the function:
y = 2 for x positive but less than or equal to 1 and y = 0 otherwise.
If you want to use a standard notation, you can just write U(t) which is 0 for t<0, otherwise 1.

Not sure what this dot and double dot refers to. The first thing that comes to mind are the first and second derivatives with respect to time.
Even the multivalued (at edges) square wave is not perfectly reproduced. There is always a small "over shoot." This is Gibb's Effect ( or phenomena ) seen below:
OK but the Fourier transform of a step function is an idealization. The Gibbs effect converges to a pure square edge in the FT.

When responding to the OP, "What is time?", I took the approach that the "how" of time dilation requires a mechanistic explanation, and discussed the hypothesis of "gravitational wave energy density of the medium of space" as a possible solution. I'm just saying that there are mechanics in the nature of the universe that govern the "how" of the observable of time dilation. Generally accepted science has yet to establish a consensus on the mechanics, but threads like this establish the opinion that there is more to time than the mathematics of spacetime explains; a view which I see you might agree with. I think the recent discussion is in line with the OP. No need to start a new thread when the discussion is on-topic, and I'm not making the discussion about the alternatives, just that there are possible mechanics to explain time dilation.

I believe in a previous comment I said i don't believe time dilation has anything to do with the question, What is time?

That's because it's a finite approximation of an infinite series.
No. The overshoot is still there even no matter how many terms, infinity included. - The drawing I showed had 125 terms. All terms for the square wave have a odd number of half cycles.

The first, a single half cycle, by its self overshoot in the middle, then the next has a pair of over shoot with the two peaks closer to the edges and its dip in the middle removes the first term's central over shoot. The next with five peaks kills the second's two overshoot and brings the center back up, but never mind that. - As the point is every higher odd frequency term removes the prior term's two near the edges and replaces them with two new overshoots still closer to the edge etc. I.e. the 100,001 frequency term has two very near the edges over shoots but it did remove the two near edges over shoots made by the 99,999 term etc to infininity. Never is the Gibbs over shoot removed or even made zero. - it just keeps moving nearer to the edges. As you can see in the 125 terms it actually grow taller for each of the most recent terms but it is bounded as all the highest terms are with less amplitude than earlier ones.

You and also understand this from a slope point of view because at the edges, the square wave slope is infinite. Consider the edge at x = o (part of the vertical line x = 0.) No matter how high the frequency of the sin wave component are, as the leave the x -axis, they curve toward positive x (away from the line x = 0). To come closer to the line x = 0, the amplitude of the highest frequency is ALWAYS GREATER than the flat top of the square wave and ALL the lower frequency components add a little more to that highest frequency component's over shoot. but this positive sum of contributions does converge to a well known limit. That is forever true no mater how many terms are in the Fourier sum. I.e. for some very small x the first peak of the highest frequency sin wave it closer to the edge but its amplitude is always greater than the top of the square wave in amplitude.

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I took it you meant the step function (integral of the impulse).
An impulse of zero duration and infinite amplitude that can be integrated is normally called a delta function. If your "integral of the impulse" is not a delta function, The value of the integral starts at zero and grows - point being it would not produce the vertical sides of a step function.
I thought I understood but maybe not. I assumed you meant "the amplitude of a sample" commonly drawn as a dot.
I said what my function was and where the two dots were with "mathematical precision." I. e. they were:
Here is the function: y = 2 for x positive or zero but less than or equal to 1 and y = 0 otherwise.
Here are where the two dots are: (0,1) & (1,1) in Cartesian coordinates, of course.

I. e. it is like your square wave but without the vertical sides. - The slope is zero everywhere. (I made it 2 tall so could tell where the dots were with integers.)
"Step function" is clear enough.
apparently not as even with my prior effort (telling it had a discontinuity) you still seem to think I am speaking of a square wave. I even gave and repeated above an example of the type of function I spoke of. Perhaps these words will help: a function y = f(x) that has one and only one value (not multi valued I said before) for all x but is not continuous.
Not sure what you mean here but it's understood the discontinuity in a step function is an idealization.
I was speaking of a class of functions that is not multivalued and is discontinuous. - I.e. at some x the limit from lesser x is not the same a the limit from higher x. In my above example the slopes from either side were both zero but the gap was 2.
... The Gibbs effect converges to a pure square edge in the FT.
I don't think so. See my argument in reply to Trippy especially the text I made bold about the highest frequency component (even in the infinite limit) always having an over shoot, an amplitude greater than the flat top, in part to make the sin more nearly vertical as it value increase from zero; and all the lower frequency terms contribute a little to that over shoot as they need a larger x before they reach their peaks and turn downward.

Anyway, I don't want to make detour longer with discussion of the Gibbs phenomena as that was not even mentioned in my first posts noting FT is not perfect for functions with a vertical gap - The FT puts a dot in the middle of the gap. I only noticed the Gibb flaw while searching for reference to the gap dot (I had long known about it but had forgotten it). Perhaps the name of FT's falsely inserted dots will come back to me and I can find reference about it, but I'm thru with this detour, I think.

PS I see I did not reply to your example with a single vertical step at the origin - I was only going to note that it had no "natural period" so that makes FT analysis more complex.

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That's because it's a finite approximation of an infinite series.
Exactly. And we can call that truncation an error similar to computational roundoff. In practice, as with signals, the presence of reactances in transmission lines or circuits inexactly matched will produce errors so many decimal places to the right - effectively the same as truncation or roundoff error, even though not exactly the same same value of error. In other words any error will produce some variation of the Gibbs effect.

And then the point is that as you extend the series out to infinity, the ringing and overshoot from Gibbs converges to zero and the edge squares up. In a practical system this amounts to carefully matching impedances which today is helped by design simulators. Otherwise high speed CPUs would not be feasible as the Gibbs effect would cause all kinds of false triggers. Pretty amazing when you realize the fundamental (first term in the series) is zipping along in the radar band!

An impulse of zero duration and infinite amplitude that can be integrated is normally called a delta function.
Yes but the delta function commonly used in Fourier analysis is normalized to unity.
If your "integral of the impulse" is not a delta function, The value of the integral starts at zero and grows - point being it would not produce the vertical sides of a step function.
The convention is that both the impulse and step functions are normalized to unity.
I said what my function was and where the two dots were with "mathematical precision." I. e. they were:
Here is the function: y = 2 for x positive or zero but less than or equal to 1 and y = 0 otherwise.
Here are where the two dots are: (0,1) & (1,1) in Cartesian coordinates, of course.
OK by "dots" you mean "points". I will PM you. I have a theory about something.

I. e. it is like your square wave but without the vertical sides. - The slope is zero everywhere. (I made it 2 tall so could tell where the dots were with integers.) apparently not as even with my prior effort (telling it had a discontinuity) you still seem to think I am speaking of a square wave. I even gave and repeated above an example of the type of function I spoke of. Perhaps these words will help: a function y = f(x) that has one and only one (not multi valued I said before) for all x but is not continuous.I was speaking of a class of functions that is not multivalued and is discontinuous. - I.e. at some x the limit from lesser x is not the same a the limit from higher x. In my above example the slopes from either side were both zero but the gap was 2.I don't think so. See my argument in reply to Trippy especially the text I made bold about the highest frequency component (even in the infinite limit) always having an over shoot, an amplitude greater than the flat top, in part to make the sin more nearly vertical as it value increase from zero; and all the lower frequency terms contribute a little to that over shoot as they need a larger x before they reach their peaks and turn downward.
By convention we call the slopes "rise time". Let me defer discussion here until we have our chat by PM.

Anyway, I don't want to make detour longer with discussion of the Gibbs phenomena as that was not even mentioned in my first posts noting FT is not perfect for functions with a vertical gap - The FT puts a dot in the middle of the gap. I only noticed the Gibb flaw while searching for reference to the gap dot (I had long known about it but had forgotten it). Perhaps the name of FT's falsely inserted dots will come back to me and I can find reference about it. but I'm thru with this detour, I think.
Probably this will clear up shortly.
PS I see I did not reply to your example with a single vertical step at the origin - I was only going to note that it had no "natural period" so that makes FT analysis more complex.

Not really once you realize it's the expansion of sin(x)/x.

Here is an example from a typical table of Fourier transforms:

As for the detour, I think this a good segway into frequency domain and time domain transformation, something that eludes the cranks here yet offers another kind of insight. Namely, in the freq domain the spectrum can remain constant and yet we can measure time elapsing in the time domain.

Also note, the photon wavelet has the shape of a sinc(t) function. This transforms to a step function in the frequency domain. Something which should put the cranks back in their cribs with their pacifiers in their mouths.

Thank you for taking the time to post that. I will spend some time on it; off hand it is reminiscent of the relativity of simultaneity. But then it looks like you apply the "true only" zone to it. Can't wait to get into it.
Yes, that is my conclusion; you have taken Einstein's explanation of the relativity of simultaneity, and overlaid it with MD's box and the "true only" zone. Am I right that it extends the "true only" zone to include the consensus view of the relativity of simultaneity?

Farsight, in a mathematical equation t is variable, its value is unassigned.
No problem.

A second has a fixed meaning based on the observer's frame of reference.
Only the observer's frame of reference is an abstract thing. It isn't something that actually exists. It isn't real. However what is real is a NIST optical clock, and it's very accurate. So accurate that when you start with two identical clocks and lower one by 30cm, it goes detectably slower. The two clocks no longer stay synchronised.

Only in your imagination can you measure the duration of a second in any frame of reference other than the one you occupy. The fact that measurements made in different frames of reference don't agree and are explained as time dilation, is not a proof that the second has anything other than a single meaning.
Rubbish! The second has a greater duration in a time-dilated environment. Hence the GPS clock adjustment. You might think it has the same duration because your measurement doesn't appear to change, but that's a tautology. It's like measuring the length of your shadow with the shadow of your stick.

BTW it is that same variable t mentioned in the first sentence that makes GR dynamic and changing. You can wrongly, as you do, assume that t always has a fixed value, and space-time is a frozen slice of reality, but the truth is that it is variable and thus imparts to space-time and GR a dynamic and changing character. Sadly it does take, some imagination and a bit more understanding of the math, to comprehend the added dynamic aspect, the variable t provides. You seem to have been repeating telling yourself the same thing for so long, that you have lost the intelectual flexibility to see your own mistake(s).
What mistake?

Only the observer's frame of reference is an abstract thing. It isn't something that actually exists. It isn't real.

And the second is any different? Hand me a second would you! It is an abstract concept we use to describe a previously agreed upon standard of change.

However what is real is a NIST optical clock, and it's very accurate. So accurate that when you start with two identical clocks and lower one by 30cm, it goes detectably slower. The two clocks no longer stay synchronised.

And the two NIST clocks you keep mentioning are not just 30 cm difference in altitude, they are in different labs. Add to that that even if they were side by side there is no certainty they would remain synchronized. How many atomic clocks are compared to arrive at our current time standard.., and where are they located? And do they all agree.., or is there an average that is drawn from the group as a whole? Look it up.

Rubbish! The second has a greater duration in a time-dilated environment. Hence the GPS clock adjustment. You might think it has the same duration because your measurement doesn't appear to change, but that's a tautology. It's like measuring the length of your shadow with the shadow of your stick.

Your comment above is a belief, even at this point bordering on a theology! You provide no proof aside from your own conclusions.., which run counter to general consensus. And talking about tautologies, look back over your own posts to see how often and for how long you have been saying the same thing. Just because you convince yourself with a religious repition, does not mean you have convinced anyone else or proven anything.

What mistake?

I am not going to go back and dig out the posts but you have repeatedly asserted that space-time was not dynamic. That it repesents only frozen slices of reality. While it is true you can plot a light cone that depicts reality as it exists for an instant where t is given a specific value, that is not the whole of GR or space-time. It is one application. What you are doing is like looking at a cube face on to one side and declaring that it is nothing more than a 2D square.

Not seeing the bigger picture is a mistake, or perhaps a disability, at this point. You have told yourself the same thing for so long that maybe you cannot see anything else.

... a NIST optical clock, and it's very accurate. So accurate that when you start with two identical clocks and lower one by 30cm, it goes detectably slower. ...
An interesting observation and I think true, (but don't vouch for 30cm being enough).

It like 200+ years go when the second was defined by pendulum clocks - They did not specify the gravity strength they were to run in. Do we do that now? - Tell what gravity strength is associated with the number of cycles that are a second?

Only the observer's frame of reference is an abstract thing. It isn't something that actually exists. It isn't real. However what is real is a NIST optical clock, and it's very accurate. So accurate that when you start with two identical clocks and lower one by 30cm, it goes detectably slower. The two clocks no longer stay synchronised.

No it is most certainly not. It is as real as any other FoR especially from the PoV of someone in that FoR.
Again, clocks have nothing at all to do with the reality or otherwise of time. It simply measures the progress of time.
And finally, everyone, in every FoR, will measure his clock as progressing at one second/second, and the speed of light as "c".
They are all facts and postulates of relativity.

An interesting observation and I think true, (but don't vouch for 30cm being enough).

It like 200+ years go when the second was defined by pendulum clocks - They did not specify the gravity strength they were to run in. Do we do that now? - Tell what gravity strength is associated with the number of cycles that are a second?

The 30cm difference in elevation is accurate for the reference.., but that is not the whole story. The two NIST clocks, I believe cesium clocks, but have not checked the reference in a long time, are in different labs in the same building. So they are separated by more than just 30cm in elevation.

The second was defined and then the number of cycles corresponding to the ground state frequency of a cesium atom, whose environment is precisely defined, was matched to the second. That resulted in a more accurate clock. More ticks to each second...

The difficulty is that there are many many variables that can affect the clock's environment and the cesium atom's ground state frequency. Location in a gravity well being only one. Temperature, magnetic and electromagnetic fields play a part.... The mechanisms used to calibrate two side by side cesium clocks may not even incorporate identical settings to achieve the same calibration.., and keeping side by side atomic clocks synchronized is a dynamic process, that is it is an ongoing process. They cannot just be synchronized and expected to agree for eternity.

So the quetsion in the last sentence quoted above, is location in a gravity well or strength of gravitation is only one variable that affects the accuracy of an atomic clock, the frequency or cycles per second...

Still the NIST reference did add data to support the prediction that a clock's location in a gravity well affects its rate. Since this thread began with the question, What is time? And that question has existed and been disputed without resolution, since the beginning of time (pun intended), the NIST clock discussion adds little clarity to the question in the OP.

Only the observer's frame of reference is an abstract thing.
The opposite is true. The only things not abstracted are the things observed in the observation frame. The rest are called apparent, a term John Duffield avoids like the plague.

The things which are abstracted include "coordinate speed of light", something Farsight claims is real. Of course, he is confusing "abstracted" and "apparent", and probably deliberately avoiding "apparent".

Also note, John Duffield is probably the puppeteer who occasionally launches Sock Puppet Army of Zealots (SPAZ) attacks on the site. Witness the occasional crank who attests to coordinate speed as real, instead of (curses!) apparent.

It isn't something that actually exists. It isn't real.
The only things that are real are those in the observation frame. The rest are apparent, and entirely subject to the angle of the Lorentz rotation. (Curses! Foiled again!)

However what is real is a NIST optical clock,
A lot of things are real, but any device at NIST is only apparent to a relative observer.

They aren't "optical". That's just a ploy to insinuate c into this, leading to Duffield's lifelong war on science, beginning with discrediting the constancy of c, probably to shore up the belief that radiometric dating is wrong, and Ussher was right. Nothing could massacre the sum of human knowledge more than this intellectual catastrophe (speaking of catastrophism).

and it's very accurate. So accurate that when you start with two identical clocks and lower one by 30cm, it goes detectably slower. The two clocks no longer stay synchronised.
That's equivocation, in avoidance of the physical phenomenon demonstrated at NIST several years ago, namely, that due to the Lorentz rotation that occurs between reference frames of NIST observers, and reference frames of each ion detected in each clock, whose chambers are at different elevations, the observers can measure time dilation and contraction, simply by watching the readouts of each instrument diverge.

Oh, maybe he meant "ocular", not "optical". I borrowed that from Thomas Paine's characterization of the "ocular demonstration" of the Ascension, reported nowhere else in Jerusalem other than by the "proxies" in the story, leading Thomas to ask for a "manual demonstration".

Note how Duffield's demands for particular kinds of proofs (usually absurd) seem to caricature the doubtful Thomas.

Rubbish! The second has a greater duration in a time-dilated environment.

Speaking of rubbish, another poster, probably one of Farsight's SPAZzes, also uses "environment" as a euphemism for "reference frame". Probably they think they will burn in Hell for an extra 30 days every time they say "reference frame" or "c is constant in all frames", and probably for 4.7 billion years, if they say "radiometric dating is accurate".

Geolocation is a calculation done in the receiver. Received sat. time tags are corrected for the dilation and contraction of SR/GR, that is, mapping the Lorentz rotation through its inverse, so that all time differences are calculated in a common frame. This type of correction is equivalent to any standard calibration procedure done routinely in all empirical measurements which employ instrumentation.

To argue otherwise is to argue that x + T[x] is valid, and to deny all of measurement science. Speaking of rubbish, that is.

You might think it has the same duration because your measurement doesn't appear to change, but that's a tautology.
Farsight likes euphemisms that imply that science is fundamentally broken. It's part of the SPAZ attackers game plan, to pepper these threads with propaganda plausible to sheeple.

And note the irony of Farsight attacking tautology.
So close yet so far, Farsight. Indeed, the Lorentz rotation, projected onto the observation plane, is best characterized as a shadow. Shamefully, even Ptolemy (2nd c.) knew that the differences in lengths of shadows had to be due to a rotation, and realized that not only was the Earth round, but he could estimate its radius, by measuring angles of shadows at two different latitudes, knowing the distance between them.

Alas, Ptolemy is not here to haunt Farsight into remorse for his crimes against academia, the way the three ghosts haunted Scrooge in A Christmas Carol.

What mistake?
"Not passing freshman physics!" shrieks the ghost of Ptolemy Present, rattling his chains and leaving Duffield's hair standing on end. (Or stubble or whatever.)

I guess that leaves Duffield back in the 1st c, maybe earlier, although I suspect he is less cynical about Aristotle's Prime Mover.

I would be interested in anyone who could define time scientifically ?

Time is simply that which separates you and I by a period of 13.83 billion years from the BB.
Your irrelevant quotes are from an irrelevant, obscure book, by irrelevant obscure people, from an age of obscurity, and means SFA in a science forum.

Yes, that is my conclusion; you have taken Einstein's explanation of the relativity of simultaneity, and overlaid it with MD's box and the "true only" zone. Am I right that it extends the "true only" zone to include the consensus view of the relativity of simultaneity?

No. Einstein says it always takes .5 seconds to the z receiver. Einstein can't tell you the velocity of the box. Einstein gets the times all wrong. Einstein is just plain wrong in a no wrong zone.

No. Einstein says it always takes .5 seconds to the z receiver. Einstein can't tell you the velocity of the box. Einstein gets the times all wrong. Einstein is just plain wrong in a no wrong zone.
If I understand you correctly, you are talking about the MD's box diagram where Einstein would consider the light source to be moving with the box instead of remaining in a fixed location relative to the motion of the box, and thus Einstein would get .5 seconds to the z receiver as opposed to your time calculation of .65 seconds?

I guess I got the implications wrong in regard to the second diagram that combined train/embankment/MD's box, but still, do you confirm that the second diagram applies the "true only" zone to Einstein's thought experiment about the embankment and the train, with the difference being that the times you calculate are different from the times that Einstein's perspective would produce? Yours being corrections to the special relativity calculations which would be considered erroneous in the true only zone?

If I understand you correctly, you are talking about the MD's box diagram where Einstein would consider the light source to be moving with the box instead of remaining in a fixed location relative to the motion of the box, and thus Einstein would get .5 seconds to the z receiver as opposed to your time calculation?

Yes, I'm talking about MD's Box and how it takes .65 seconds for the light to reach the z receiver. The source is fixed to the center of the box at all times. It travels with the box and stays at the center of the box. It probably has welded beams holding it in place?? Anyway, Einstein also considers the source to remain at the center of the box at all times. He just stands there and looks at the source at the center of the box, and then he sends a light sphere and claims it takes .5 seconds for the sphere to reach all receivers at the same time, t=.5 seconds. He doesn't understand it can take any different time to reach the receivers because he doesn't understand the box can have a velocity in space. According to Einstein, and everyone else in the history of the world, it is impossible to tell if the box is in motion or not if the motion is constant (not accelerating).

I am claiming they are all FOS! I claim MD's Box! That the box has a velocity and it takes different times to reach the receivers, depending on the velocity of the box.

Again I remind you that this is the 2D rendition. Imagine a 3 dimensional real sphere expanding. There is 3 axis, x, y, and z. The sphere is not a cube, and as it grows the rounded part has to find the time of an object at the exact time at the EXACT location, like in MD's box. Can you even imagine that, that the 3D sphere is expanding its radius, and an object could be traveling away from (0,0,0) not along any axis, but as it damn well pleases at any velocity. Things get pretty nasty pretty quick, let me tell ya!!

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Yes, I'm talking about MD's Box and how it takes .65 seconds for the light to reach the z receiver. The source is fixed to the center of the box at all times. It travels with the box and stays at the center of the box. It probably has welded beams holding it in place?? Anyway, Einstein also considers the source to remain at the center of the box at all times. He just stands there and looks at the source at the center of the box, and then he sends a light sphere and claims it takes .5 seconds for the sphere to reach all receivers at the same time, t=.5 seconds. He doesn't understand it can take any different time to reach the receivers because he doesn't understand the box can have a velocity in space. According to Einstein, and everyone else in the history of the world, it is impossible to tell if the box is in motion or not if the motion is constant (not accelerating).

I am claiming they are all FOS! I claim MD's Box! That the box has a velocity and it takes different times to reach the receivers, depending on the velocity of the box.

Again I remind you that this is the 2D rendition. Imagine a 3 dimension real sphere expanding. There is 3 axis, x, y, and z. The sphere is not a cube, and as it grows the rounded part has to find the time of an object at the exact time at the EXACT location, like in MD's box. Can you even imagine that, that the 3D sphere is expanding its radius, and an object could be traveling away from (0,0,0) not along any axis, but as it damn well pleases at any velocity. Things get pretty nasty pretty quick, let me tell ya!!
It makes me dizzy just thinking about it, lol, but thanks for the additional explanation; food for thought.

If Farsight were right and space-time represented only a single slice or fixed instant
Huh? Spacetime represents space at all times. That's why there's no motion in spacetime.

And the second is any different? Hand me a second would you! It is an abstract concept we use to describe a previously agreed upon standard of change.
We all know what a second is. It's nowadays defined as the duration of 9192631770 periods of radiation. Only it varies with altitude. If it didn't, time dilation wouldn't happen.

And the two NIST clocks you keep mentioning are not just 30 cm difference in altitude, they are in different labs. Add to that that even if they were side by side there is no certainty they would remain synchronized. How many atomic clocks are compared to arrive at our current time standard.., and where are they located? And do they all agree.., or is there an average that is drawn from the group as a whole? Look it up.
Ye Gods, now you're casting aspersion on gravitational time dilation?

Your comment above is a belief, even at this point bordering on a theology! You provide no proof aside from your own conclusions.., which run counter to general consensus.
Oh geddoutofit. What I'm telling you is legit GR gravitational time dilation. Go and do your own research.

And talking about tautologies, look back over your own posts to see how often and for how long you have been saying the same thing. Just because you convince yourself with a religious repition, does not mean you have convinced anyone else or proven anything.
You're the one being religious here. Not me.

I am not going to go back and dig out the posts but you have repeatedly asserted that space-time was not dynamic. That it repesents only frozen slices of reality. While it is true you can plot a light cone that depicts reality as it exists for an instant where t is given a specific value, that is not the whole of GR or space-time. It is one application. What you are doing is like looking at a cube face on to one side and declaring that it is nothing more than a 2D square.
This is getting surreal. You plot a worldline in spacetime. The worldline represents the motion of a body through space over time. If the worldline is straight up the body isn't moving through space. If the worldline is slanted the body is moving through space. But in either case the body isn't moving through spacetime.

Not seeing the bigger picture is a mistake, or perhaps a disability, at this point. You have told yourself the same thing for so long that maybe you cannot see anything else.
Again, go and do your own research to investigate whether I've told you the correct physics. You will find that I have. And that's you've been dismissing it in favour of popscience misunderstanding.

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