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.
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:
I took it you meant the step function (integral of the impulse).By "step" I was referring to a discontinuity.
I thought I understood but maybe not. I assumed you meant "the amplitude of a sample" commonly drawn as a dot.- Sort of obvious as I said the Fourier transform would have a dot at the mid point,
"Step function" is clear enough.but I could had said: "The FT of a function which is no where multivalued
If you say "square wave" esp in connection with the FT it's understood to be the idealization.(as is the square wave at it edges)
Not sure what you mean here but it's understood the discontinuity in a step function is an idealization.and discontinuous is not perfect.
If you want to use a standard notation, you can just write U(t) which is 0 for t<0, otherwise 1.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.
OK but the Fourier transform of a step function is an idealization. The Gibbs effect converges to a pure square edge in the FT.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:
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.
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.That's because it's a finite approximation of an infinite series.
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 took it you meant the step function (integral of the impulse).
I said what my function was and where the two dots were with "mathematical precision." I. e. they were:I thought I understood but maybe not. I assumed you meant "the amplitude of a sample" commonly drawn as a dot.
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."Step function" is clear enough.
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.Not sure what you mean here but it's understood the discontinuity in a step function is an idealization.
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.... The Gibbs effect converges to a pure square edge in the FT.
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.That's because it's a finite approximation of an infinite series.
Yes but the delta function commonly used in Fourier analysis is normalized to unity.An impulse of zero duration and infinite amplitude that can be integrated is normally called a delta function.
The convention is that both the impulse and step functions are 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.
OK by "dots" you mean "points". I will PM you. I have a theory about something.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.
By convention we call the slopes "rise time". Let me defer discussion here until we have our chat by PM.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.
Probably this will clear up shortly.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.
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?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.
No problem.Farsight, in a mathematical equation t is variable, its value is unassigned.
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.A second has a fixed meaning based on the observer's frame of reference.
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.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.
What mistake?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).
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.
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.
What mistake?
An interesting observation and I think true, (but don't vouch for 30cm being enough).... 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. ...
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.
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 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.Only the observer's frame of reference is an abstract thing.
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!)It isn't something that actually exists. It isn't real.
A lot of things are real, but any device at NIST is only apparent to a relative observer.However what is real is a NIST optical clock,
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.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.
Rubbish! The second has a greater duration in a time-dilated environment.
The clocks are not adjusted.Hence the GPS clock adjustment.
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.You might think it has the same duration because your measurement doesn't appear to change, but that's a 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.It's like measuring the length of your shadow with the shadow of your stick.
"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.)What mistake?
I would be interested in anyone who could define time scientifically ?
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?
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?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?
It makes me dizzy just thinking about it, lol, but thanks for the additional explanation; food for thought.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!!
Huh? Spacetime represents space at all times. That's why there's no motion in spacetime.If Farsight were right and space-time represented only a single slice or fixed instant
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 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.
Ye Gods, now you're casting aspersion on gravitational time dilation?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.
Oh geddoutofit. What I'm telling you is legit GR gravitational time dilation. Go and do your own research.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.
You're the one being religious here. Not me.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.
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.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.
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.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.