steveupson

I'm pretty sure that I haven't overcomplicated anything, and I'm pretty sure that I haven't hijacked anything, and if I do have the lack of understanding that you are so sure that I have, well, then that is specifically why I am here asking questions. The image shows what I've been saying. How is the math over complicating things? If there's a simpler explanation of either the math that has been presented or of the question that was asked in the OP then share it with the rest of us. If I've said anything that contradicts your calculus methodology then please correct me because I don't know what it is that I said that you believe to be wrong. Sure, I disagree with the way we've always done things, that's correct. But can't we put on our grown-up pants and discuss why the math that I've presented is simply dismissed in favor of the old "we do it this way because we've always done it this way" rationale? Have you even looked at the math before commenting on it? I think not. We generally have a low opinion of people who ignore the math in favor of pablum. Don't be one of those people. What, specifically, have I said - EVER - that you take issue with? If members do believe this to be a thread hijack then vote this post down. If you're interested in this discussion then please vote it up.

To recap: The arc length of a velocity vs time (or position vs time) graph is a line (minimum length) if the underlying data is a constant function and will not be a line if the underlying data is not a constant function. If the variation in the function is cyclic then the arc length is proportional to frequency and amplitude of the variation. The amplitude is simply a scale factor, so the frequency component must be relevant part. For a cyclic (sinusoidal) function, the frequency is simply the angular velocity. Angular velocity is the change in angular position over time. Angular position is direction. This quantity (arc length of a velocity vs time graph) has a base quantity (or dimension) of direction. (Yes, I do understand that this is not in the wiki for this subject, but let's try and be scientists here and look at the actual issues. The questions that are raised in the OP will still exist whatever my behavior is and no matter what I have to say about it. If the math that is being presented is incorrect, please try and focus on fixing the math instead of dwelling on trying to fix the author of this post.) We know that direction is always expressed as a ratio between two lengths (xyz vectors or diameter to circumference) so it’s natural to assume that when we say that the base quantity is direction that we mean that the dimension is length. But this is wrong because base quantities (or loosely, a dimension) must be in units that are numbers, and not a ratio. What is missing here is the number or quantity that would express direction as a number and not as a ratio. There is way to do this but it does involve doing some math in order to understand how it is done. There isn’t any intuitive way that I know of that would allow a person to quantify direction as a base unit without doing math. Fortunately, the math is simply trigonometry and involves solving some triangles. The model can be viewed as an interesting math problem. The truly competent folks seem to view it this way, having made an honest effort to find a solution before expressing an opinion. Others, who also seem to be extremely competent but not quite as careful, complain that the problem is so simple and trivial that it isn’t worth the time or effort to try and solve. The solution that we’ve come up with gives two equations that are a parameterization of a multivariable function. At least that is what I’ve been told. The two equations are: [latex] \cot\alpha = \cos\upsilon\tan\frac{\phi}{2 } [/latex] [latex] \sin\frac{\lambda}{2 } = \sin\frac{\phi}{2 }\sin{\upsilon} [/latex] The function [latex] \alpha=f(\lambda) [/latex] turns out to be: [latex] \alpha={\cot}^{-1 }(\cos\upsilon\tan{\sin}^{-1}(\frac{\sin\frac{\lambda}{ 2}}{ \sin\upsilon})) [/latex] This smooth function approaches a sine curve when [latex] \upsilon\to0 [/latex], and that approaches a hyperbola when [latex] \upsilon\to\frac{\pi}{2} [/latex] , and allows a way to assign a meaningful value to direction, rather that simply keep considering it to be a simple ratio (which it is - but it's also more, much more, than that.) More about the interesting math problem is here: https://forum.cosmoquest.org/showthread.php?161772-New-Math-Function-Redux (the solution attempts begin at post #164)

Galileo

Try and keep up. The OP asked about the significance of the arc length in a velocity vs time or displacement vs time graph. At least two of our members (and perhaps yourself) are discussing how the arc length would be determined. I asked them a question because they seem to know stuff. Of course to me, the better - more interesting - question is what do you have when you know the arc length. That’s what I interpret the OP to be asking. I’ve also been informed by the moderation staff (right on que) that my views are not to be considered for discussion. Maybe it’s because ya’ll can’t keep up. That seems to be your problem. I would suggest that when you do figure out what you’re doing and what you’re talking about that you consider what dimension the arc length is in. But oh wait, we can’t go there, can we? Even when another member asks this identical question.

If the variation is cyclic in nature then wouldn't it be a candidate for Fourier analysis, and if it isn't cyclic then wouldn't it be a candidate for fractal analysis?

Of course. This should have read velocity of light observed as image A or B.

I think my questions have all been answered. Thank everyone for their participation.

Yes. Like I said before, you're probably right and I'm probably wrong. I guess I'm wrong about wanting to learn, too. You probably know much better about that than I do. In any event, can you plot the velocity of image A vs time and image B vs time? Or even better, can you plot the displacement of image A vs time and image B vs time? Assume that the twin quasar is 7,800,000,000 ly distant from earth. Since the displacement is the same, and the magnitude of the velocity is the same, by your methods the the two graphs should be identical, shouldn't they? Rather than respond to the issues that have been raised you choose to offer opinions about me. What would you call it?

I must have misunderstood. I thought your post was about me. Is there some other content that I missed?

You're probably right and I'm probably wrong. In any case, can you plot the velocity of image A vs time and image B vs time? Or even better, can you plot the displacement of image A vs time and image B vs time? Assume that the twin quasar is 7,800,000,000 ly distant from earth. Since the displacement is the same, and the magnitude of the velocity is the same, by your methods the the two graphs should be identical, shouldn't they? Teach me. I don't understand your method. I do want to learn. How I think about this doesn't really have any effect on the physics. These ad hominem arguments from you are becoming wearisome.

From the OP, specifically we're talking about a graph of velocity vs time or position vs time. I think it's appropriate to assume spacetime as part of the discussion. on edit>>> An example would be that the plot of position or velocity vs time of a race car around a circular track will be a sine wave. The plots of a race car down a drag strip will be a parabola. When speed is the constant (as with the speed of light) then the plots of the drag strip are a line and they are the plots that represent this constant.

In spacetime that is not curved (as with Euclidean 3-space) the two definitions are identical. It's only when we consider curved spacetime that the issue arises. The problem with speed being the magnitude of velocity is that in curved spacetime the velocity of light can be any value less than the speed of light. The two are not the same in nature (curved spacetime) and we accept speed as the constant, not velocity, by convention. In the twin quasar example we have one change of position (or displacement) and three different times. The magnitude of the velocity for the straight-line or geodesic path is [latex]c[/latex] which is the rate of change of position of light. The magnitude of the velocity for image B is different because the rate is [latex]c- n_1[/latex], where [latex]n_1[/latex] is the difference in velocities caused by the time difference due to path difference. The magnitude for image A is [latex]c - (n_1 + n_2)[/latex] where [latex]n_2[/latex] is the change in velocity due to the 1.1 year delay caused by the distance traveled for image B being 1.1 ly further than that traveled by image A. Or, in order for the velocities to be identical for all three times, the displacement between the quasar and earth must be different for each path. That doesn't really fit the definitions that we're using. On a similar note, there are other derived units that are ambiguous in the same way because they use [latex]l^3[/latex] as a volume. Once again, this loses any meaning at all in curved spacetime for similar reasons. In curved space [latex]l^3[/latex] becomes ambiguous because length, once again, is not a constant. Yes. The velocity of light in curved spacetime is not a constant. The speed of light is a constant and is represented by the shortest length. I'm using the term geodesic and I hope that is a proper use of the term.

Yes, true, but vectors are ratios, not numbers. And there's a special treatment in this particular case of speed. It isn't like any of the other vectors. The x,y,z components of a vector contain a ratio that describes a direction in Euclidean 3-space. This ratio is pretty much meaningless in spacetime. The reason it's meaningless is that the speed of light is constant, and therefore length isn't a constant (it's simply a base quantity) and it varies in a manner that is inversely proportional to time. "Vectors have magnitude and direction, scalars only have magnitude. The fact that magnitude occurs for both scalars and vectors can lead to some confusion. There are some quantities, like speed, which have very special definitions for scientists. By definition, speed is the scalar magnitude of a velocity vector." https://www.grc.nasa.gov/www/k-12/airplane/vectors.html We also have another method of addressing direction: angles. The same is true for this treatment. An angle is a ratio that exists between the diameter and circumference of a circle. Once again direction is expressed as a ratio and not as a quantity. Spherical excess expresses direction as a quantity, a number, not as a ratio. The relationship between a flat plane (Euclidean 2-space) and the surface of a sphere can be expressed as a real quantity. By this I mean that it can be expressed as a scalar value. This is a real quantity, or at least I believe that it meets the technical definition of one since it's a number and not a ratio. If spherical excess isn't a quantity then what is it? There's another new way to express the quantity that manifests spherical excess. This new method permits the curvature of spacetime to be linearized. If we call the traditional method of quantifying spherical excess the linear method then we could refer to the new method as the circular method. It produces more information than the traditional method. Basically, what it shows is that space that isn't curved (no spherical excess) is dominated by the sine function while space with maximum curvature (maximum spherical excess) is dominated by the hyperbolic function. By the term linearized I mean that since a smooth function exists between a sine and a hyperbola, there is a way to scale the curvature. Again, explain how this gets resolved in the twin quasar question. Thegravitational lensing associated with the twin quasar shows this to be incorrect to a certainty. "30 years of observation made it clear that image A of the quasar reaches earth about 14 months earlier than the corresponding image B, resulting in a difference of path length of 1.1 ly." https://en.wikipedia...iki/Twin_Quasar By definition, the light from A has a higher velocity than the light from B, even though they have the same speed. There should be three separate values with different graphs; velocity of light for image A, velocity of light for image B, and a third velocity c for speed of light (based on the geodesic between the quasar and the earth.) Yes. It should account somehow for the path taken, but it really doesn't. I think we can agree that the path can go in different directions, we disagree on what that means. My understanding is that spacetime is a combination of two different things, time and space. I also understand that space is also a combination of two different things, direction and distance. Spacetime has three base quantities, time, length, and direction. This is different than saying that spactime has two base quantities, time and length and the ratio between lengths.

Right. Velocity x time produces a different result than speed x time. I raised this question (I think it's the same question) in the dimensional analysis thread. I was told there that even though the two things are different mathematically, they have the same dimension (I think that's what I was told.) The mainstream view on this question is defined that way, but it isn't explained why, at least not anywhere that I've been able to find. It doesn't make sense mathematically or conceptually. There is an alternative view that does make sense, both mathematically and conceptually. The way it should work is that since velocity is displacement over time, instead of length (arc-length in this case) over time, then there should be a velocity time graph where displacement instead of length is the area under the curve. This difference isn't as subtle as it sounds. In order to quantify displacement we have to know direction because displacement is length x direction. I can't find any evidence that anyone has seriously proposed considering direction as a base quantity on a par with length or time. This looks like a glaring omission. We know that the linear expression of spherical excess (E = A + B + C - π) isolates a quantity, and yet some insist that this quantity that is being isolated isn't a real quantity. Hopefully I won't receive bullying from the staff for publicly talking to you about this. I've been warned many times that this is a taboo subject, but you did ask. I think the question being asked publicly by another member should be grounds for me to offer a comment, even though I've been previously banned from articulating my opinion on this subject.

Yes, correct, and this time difference is why the two velocities are different.

Why does the light from A and B arrive at different times if the direction is dimensionless?

Maybe scalars and vectors are interchangeable when comes to dimensional analysis. Is that what you're saying? It doesn't make any sense at all to me. But in any case, we're talking about c, the speed of light. It's a constant whereas the velocity of light is not. The gravitational lensing associated with the twin quasar shows this to a certainty. By definition, the light from A has a higher velocity than the light from B, even though they have the same speed.

I don't see how this can be correct. Shouldn't it read: "c" is a speed, which is measured in units of meters/second? It doesn't make sense that velocity and speed have identical dimensions. Does anyone know the definition of "dimension" that would quantify velocity and speed as being the same dimension? I don't understand how this can ever work.

I think there's a lot more luck involved than anything else. Most good problem solvers try many different approaches while trying to ferret out a likely solution. The more complicated the problem, the less likely it is that a solution will be found. That's if the problem even has a solution. "Men occasionally stumble over the truth, but most of them pick themselves up and hurry off as if nothing ever happened." - Winston S. Churchill

It would fall straight down. This is the reason that the SR71 Blackbird wasn't armed. It really couldn't aim and fire a weapon effectively while moving at Mach-3. As an amusing aside, North Korea fired a surface to air missile at one. That really made no sense unless they figured that the aircraft might turn around and go back in order to intercept it. On edit >>> North Korea

Yes, of course. This is even more support for the thesis than any other proof that's been offered. Direction must contain information in order for this to be true. In the case of the skew axis, direction encodes useful information, thereby making the calculation more manageable.

That's a very good point. It helps to illustrate how a huge variety of different kinds of information can be represented this way. That's sort of the whole point behind defining direction as a base quantity. It can represent a lot more information than what we would normally ascribe to it doing it the simple way.

Thanks Mordred. Yes, that is the understanding of how the math works in the conventional scheme of things. Since each of these degrees of freedom that is being added is orthogonal, then their direction is axiomatic. It's the same (mathematically) as adding more length axes that are orthogonal to the normal three. That's very different than what I was talking about. Once again, the terminology that we're stuck with isn't very specific. Your use of degrees of freedom has a meaning that is much more specific than just using the term dimensions. The point I was making is that time isn't a length at all. It does add a degree of freedom, but not in the same way that adding another orthogonal length does. They are two distinctly different processes, both mathematically and conceptually. Time is added to 3-dimensional Euclidean space by a different method, specifically we're talking about Minkowski space. Also, direction isn't a length at all, either. It comes to the party wearing its own ensemble, too, just like time does. Because of the mathematical treatment we use in order to add these additional degrees of freedom, it's difficult to visualize exactly what is happening when we start piling up more and more orthogonal axes. Because of the way we do the math, length and direction get conflated in a very subtle fashion (it has to be extremely subtle because it hasn't been recognized until now.) If we zoom in on this subtle conflation then we can actually separate direction completely from the base quantity of length. When we do that, direction becomes another base quantity. In case you haven't understood what I've been saying all along (which I'm fairly certain no one has understood completely) let me try and say it once more, with some clarity this time. We currently only express direction or orientation as a relationship between two (or more) lengths. It never stands alone, without being associated with at least two lengths. These can be perpendicular xyz lengths or they can be the diameter and circumference of a circle. In either case we express direction by explicitly defining two lengths. There's another method for expressing direction, one that doesn't rely on any lengths at all. It expresses direction based on the relationship between three orthogonal objects. This relationship only occurs with three orthogonal dimensions. It doesn't have any existing analog since all of our expressions for direction are based on two-dimensional relationships that occur between lengths.

First, let me know if this response is off topic or if I’m hijacking your thread. My only intention here is to actually have a grown up discussion of the issue you raised (I think) in your question. I agree that there does seem to be something that is slightly off about either the accepted terminology or about the fundamental concept, or both. We can easily see that time isn’t like anything else, certainly it isn’t equivalent to a length, or, more correctly, it isn’t like two lengths that are oriented perpendicular to one another. When we start talking about dimensions there is an arbitrary explanation or understanding that has been agreed upon. As some of the experts here have already pointed out, dimensions can be different things (both conceptually and mathematically) depending on the application. The way it is mathematically expressed, Euclidean 3-space actually comprises three sets of 2D planes. It would be more meaningful to call it 2D^3 rather than 3D. That’s the reason why Euclidean 3-space has octants… 2X2X2=8. There is a deeper understanding of what a 2D plane is, though. It is two lengths perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings direction (or orientation) into the picture, along with the length which is the first dimension. So, what is direction, really? We can call it angular position, and we can call the change in angular position angular velocity, and we can call the change in angular velocity angular acceleration. In case it isn’t obvious, there is a stunning symmetry with length here, where we call the change in length speed, and the change in speed acceleration. Even more stunning is the fact that when we add direction to length we get position, when we add direction to distance we get displacement, and when we add direction to speed we get velocity. Still, even in light of all these facts, there seems to be a winning argument (for reasons no one can explain other than the old "we've always done it that way") that direction isn’t really a base quantity, like time or length. It’s supposed to be a thing that isn’t really a thing, whereas the other two are things that really are things. I think that this approach is rather arbitrary, especially when direction can be quantified as a scalar value exactly like time and length are both quantified. A more correct approach would be to identify time, length, and direction as three different dimensions. We already see time and space as separate, so all that is necessary is to recognize that space is the combination of direction and distance. It does make much more sense to view it this way because the math supports this view, and it doesn’t really support the other mainstream view, whatever that is (no one seems to be able to say exactly what the mainstream view is, only that it isn't this). I'm not a math guy either, but I do see what I think are the same issues as what you're asking about. Personally, I think that the higher dimensional spheres are one of those mathematical oddities that has very little to do with our physical universe. When we look at spacetime mathematically we see it as time orthogonal to a backdrop of hyperbolic space. I do think that this mathematical "trick" has usefulness and it does represent a major aspect of the natural universe. Everything else (string theory, etc.) must lie somewhere between these extremes.

If it were obvious we'd have nothing to discuss. That link was very helpful. I haven't studied matter, even the "normal" kind, and so I'm very much a novice on that subject. I know that I'll be making a lot of childish mistakes here and there and I hope the other members will correct me when I do. At this point everything that I look at, every post I read, every new report that comes out, seems to confirm what the basic math is saying. We look at the universe as comprising [latex]4\pi[/latex] steradians. We naturally assume, from Euclidean geometry, that all steradians are the same. This is only half true because some funny stuff starts to happen when we move from Euclidean 3space to spacetime. I'm going to try and present an additional model that is very different from the standard view, but is equally relevant mathematically. If we consider a star to be a point in the sky, then when we look at that point what we see has the same weight as what someone who is there sees when they look at us. If we look at a solid angle centered on that star then the magnitude of what we see is the same as the magnitude of what they would see if they looked at the same solid angle centered on our sun. These directions commute to one another. Each point has this reciprocal relationship with every other point. When we add a third star it sees the same solid angle as what we see when we look at it, and it sees the same solid angle as the one the first star sees when they look at them. In Euclidean 3space this is true no matter what sort of triangle the three create, it doesn't matter at all. As as we add time to the picture then everything changes. The three cannot be arranged as a classical equilateral triangle where there isn't any difference in the size of the solid angles that they all see, and there isn't any difference in the distances between them. The change occurs because there has to be at least two different lengths between any two stars that have the same angles as seen from the other one. It's one of those quirks that is born out of relativity. In this model, any reference frame can only contain one point where length commutes, where two points AB=BA. It's very much like the hairy ball thing with trying to comb a sphere. This leads to some unavoidable inequalities (at least I haven't yet found a way mathematically to avoid them.) If we see both stars in the same steradian then the quantity is different than when we see them in a solid angle that is greater than a steradian. I'm not just talking about volumes or solid angles. The actual directional density is different. This only works because time keeps these things properly oriented to one another. It would never work if spacetime didn't include relativity. If you study the function you'll see that any solid angle that we can interact with will only be [latex]2\pi[/latex] steradians, maximum. We can only see what can see us. These [latex]2\pi[/latex] steradians are also weighted in a specific manner toward a cardinal direction. Every point and particle has this [latex]2\pi[/latex] steradians relationship with every other particle, and they share a common cardinal direction. Certain things seem to make more sense once you implement this model, while others make less sense. The first thing that must be understood is that in this form, direction is a base quantity so it can't be mixed with other base quantities like length. Our normal tendency is to see the relationship between length and direction in a certain light to which we are accustomed. This model shares nothing in common with the standard geometry that I can see. In this model length doesn't have any particular orientation. The rules for combining things are different. The math can be explained, but there has to be some context first, in order to make sense of it. I should work on this post some more because it's too long and it can be improved on. Some feedback would make that task more focused, though. I'm sorry that it's become another wall of inscrutable text, but I'm unable to think of another way to do this. And yes, this is relevant to dark matter and dark energy. We need to understand that what we're actually looking at certainly isn't Euclidean 3space. At least that isn't the whole picture.