I Arguments leading to the speed of light as a dimensionless constant

[Saw]

@PeroK, you misquoted me, please edit your quote in post 48. My wink was accompanying the joke to PeterDonis where I gave him back the recommendation to go and study vector space. I introduced it to show that I was saying that in a playful way, with constructive intention, following his suggestion. But if you put it where you put it, you make me sound as patronizing, which is far from my intention.

The two concepts are, therefore, more subtly different that your naive analysis involving "shadows" would suggest.

"Naive" is an implicit assumption that the analysis is basically correct. So the burden of disproving it is displaced to you.

From a pure mathematical perspective, linear independence is an algebraic property. It depends only on the addition of vectors and multiplication by scalars. Whereas, orthogonality is an analytic property, as it depends on the inner product.

In what sense is inner product analytic? Is "analytic" here referring to calculus? If so, that will be the case when the dot product is an integral, like in Hilbert space, but not here.

Moreover, linear independence of a set of (more than two) vectors is not a case of mutual linear independence only. E.g. the vectors $(0,1), (1,1), (1,0)$ in $\mathbb R^2$ are all pairwise linearly independent, but form a linearly dependent set.

Orthogonality, on the other hand, is only a pairwise concept. A set of vectors is orthogonal if and only if every pair of vectors in the set is orthogonal.

The two concepts are, therefore, more subtly different that your naive analysis involving "shadows" would suggest.

The two vectors at the extremes are orthogonal, i.e. totally linearly independent, and this is a 2D space, so forcefully they span the whole space, including the vector in the middle, which is therefore fully redundant... I don´t see how this contradicts what I am saying, rather it looks like a confirmation thereof.

Furthermore, it is not enough if you list differences btw the two concepts, you should mention why such differences are relevant to the discussion. I brought up the idea of independence btw basis vectors just to show that it makes sense to assign the same units to T and X because, no matter if they seem independent from each other, since that does not prevent one from seeing them as ways to look at the same thing. However, if as I already mentioned, we adopted an operational method to measure T and X where these axes were only linearly independent but not orthogonal, that would not undermine the argument...

 

[Saw]

In Euclidean space, one can at least say that vectors that are orthogonal are also linearly independent (although the converse is of course not true). But in Minkowski spacetime, even that is not the case: null vectors, as I have already pointed out, are orthogonal to themselves, and they are certainly not linearly independent of themselves.

If you read my posts, you would realize that I am not saying that the vision of orthogonality as "total independence" flows into Minkowskian dot product, which is the sense in which null vectors are orthogonal to themselves (dot product with negative sign is 0). The point is that here the orthogonality concept has been generalized in a way that picks up the dot product leg, but drops the full independence leg. So it is no surprise that the Minkowskian dot product does not conform to the total independence requirement, which it does not have. I am not saying that such way of generalizing orthogonality is wrong, no doubt it is the right thing to do, I am just holding that it is a conceptual swerve.

However, what is still puzzling is that the total independence idea pops up unexpectedly in spacetime. I think that this is hard to deny if you look at the Minkowski diagram and the frame which has been painted as observer, the one usually marked as non-primed frame. If I can see well, the T axis projects no shadow over the X axis and the deltas t and x of a timelike interval look like the height and the base of a right angle triangle, embracing the hypotenuse of the proper time as measured by the primed frame. Of course, this happens because the latter is painted in another scale. But that is why I hypothesized that each frame deems itself as having perpendicular axes in the "old" sense, denies that others have this privilege and sees their lengths distorted.

I can understand that you are suspicious about this speculation, which is what it is, I can perfectly concede that. But if you also deny that the "no shadow" or "total independence" concept is what has always denoted orthogonality, until we arrive at spaces with another metric, then I am very surprised. Look at the Fourier transform! How can you explain that the basis functions of a Hilbert space are orthogonal if not by reference to the concept of total independence (the magnitude of the signal at one time-point or frequency is independent from the others)...?

Anyhow, I have convinced myself that I will not move you an inch from your positions. Please say your last word, of course, if you please, but I will not reply. The units issue has been commented to satisfaction, I have learnt quite a few things, so I thank you all for your time and interest and will not say more, unless I find some interesting material to share!

 

[Ibix]

But if you also deny that the "no shadow" or "total independence" concept is what has always denoted orthogonality

What light source is casting the shadow? Your whole "no shadow" concept hinges on the light travelling parallel to the vector, which may be arranged for any vector.

 

[Ibix]

Ibix, I said that I would not reply any more, but I think I should out of courtesy: I genuinely don't know what you mean. My "no shadow concept" has nothing to do with light, it was only this:

I was taking your "shadow" literally. If you want to be less literal: how are you doing the projection? You are doing it by some process like dropping a perpendicular from the tip of one vector to the other - but here you presuppose a notion of perpendicularity. Thus your argument is circular: your definition of two orthogonal lines relies on the existence of two orthogonal lines.

 

[Orodruin]

The two vectors at the extremes are orthogonal, i.e. totally linearly independent, and this is a 2D space, so forcefully they span the whole space, including the vector in the middle, which is therefore fully redundant... I don´t see how this contradicts what I am saying, rather it looks like a confirmation thereof.

There are many misconceptions in this paragraph.

First of all, there is no such thing as ”fully independent”. A set of vectors is either linearly independent or not. Any two of the quoted vectors are linearly independent but the set of three is not.

Second. Linear independence is not directly related to orthogonality. In fact, you do not even need an inner product to discuss linear independence. Furthermore, in Minkowski space you can have two vectors that are parallel yet linearly dependent.

You are also missing PeroK’s point of linear independence being a property of a set of vectors rather than a property of pairs of vectors.

 

[Saw]

What light source is casting the shadow? Your whole "no shadow" concept hinges on the light travelling parallel to the vector, which may be arranged for any vector.

Well, I said that I would not reply anymore, but I think I should out of courtesy, since you are asking me a question. I don't know what you are referring to. What I am saying is not a weird invention. I suppose that if I look for them, I will find many texts where they use this image to explain perpendicularity. If you project a light (an imaginary light) over an analyzed vector, perpendicularly to the analyzing vector, you get a shadow, which is the degree in which the analyzed vector shares the direction of the analyzing vector. If the shadow is zero, that means that analyzed and analyzing vectors are orthogonal (they do not share any component).

 

[Ibix]

Well, I said that I would not reply anymore, but I think I should out of courtesy, since you are asking me a question. I don't know what you are referring to. What I am saying is not a weird invention. I suppose that if I look for them, I will find many texts where they use this image to explain perpendicularity. If you project a light (an imaginary light) over an analyzed vector, perpendicularly to the analyzing vector, you get a shadow, which is the degree in which the analyzed vector shares the direction of the analyzing vector. If the shadow is zero, that means that analyzed and analyzing vectors are orthogonal (they do not share any component).

See this post:

I was taking your "shadow" literally. If you want to be less literal: how are you doing the projection? You are doing it by some process like dropping a perpendicular from the tip of one vector to the other - but here you presuppose a notion of perpendicularity. Thus your argument is circular: your definition of two orthogonal lines relies on the existence of two orthogonal lines.

 

[Saw]

Ibix, really, haven't you read this in hundreds of texts? The light source will be parallel to the analyzing vector and the light will come perpendicularly to the analyzed vector. This procedure shows the degree of directional coincidence of both vectors. I don't think there is circularity in this, but never mind!

 

[Orodruin]

Ibix, really, haven't you read this in hundreds of texts? The light source will be parallel to the analyzing vector and the light will come perpendicularly to the analyzed vector. This procedure shows the degree of directional coincidence of both vectors. I don't think there is circularity in this, but never mind!

You (or that simile) are presupposing a Euclidean space. Spacetime is Minkowski, not Euclidean, so it does not work as a mental guide in spacetime.

And no, it does not sound like it would be a good textbook simile even in Euclidean space. At least not at higher level. It may have more of an intuition point in lower level maths but even then it is flawed as presented in this thread.

 

[Ibix]

I don't think there is circularity in this

Then how do you define the word "perpendicularly" when you write "the light will come perpendicularly to the analyzed vector"? You rely on it in your definition of orthogonality.

 

[Saw]

Then how do you define the word "perpendicularly" when you write "the light will come perpendicularly to the analyzed vector"? You rely on it in your definition of orthogonality.

The shadow idea is not for defining perpendicularity, it is only for illustrating the fact that perpendicularity is independence to a higher extent: it is instead of some shadow, no shadow at all.

 

[Saw]

You (or that simile) are presupposing a Euclidean space. Spacetime is Minkowski, not Euclidean, so it does not work as a mental guide in spacetime.

And no, it does not sound like it would be a good textbook simile even in Euclidean space. At least not at higher level. It may have more of an intuition point in lower level maths but even then it is flawed as presented in this thread.

If you read my posts, you will find out that I am not positing the simile for Minkowski space, at least not for the orthogonality that is based on the dot product with negative sign, which is not the same thing, sure. I just said that the shadow simile pops up however also in Minkowski space under forms that would deserve discussion.
As to the goodness of the shadow simile in Euclidean space, I find it perfect for that domain. Does it work or not? Does it serve to tell a perpendicular vector from one that is not or not? That is all that matters. Rest is subjective: I think, I don’t think, I like, I don’t like...
But thanks indeed for the comments and bye! I said that I would not insist on these concepts that mentors are clearly rejecting and so this is over!

 

[PeterDonis]

The two vectors at the extremes are orthogonal, i.e. totally linearly independent

Not if they are null vectors.

 

[PeterDonis]

here the orthogonality concept has been generalized in a way that picks up the dot product leg, but drops the full independence leg.

I don't understand what you are talking about. This looks like a personal theory of yours. Personal theories are off limits here.

I can understand that you are suspicious about this speculation, which is what it is, I can perfectly concede that.

So it is a personal theory of yours. See above.

I have convinced myself that I will not move you an inch from your positions.

Since the "positions" that everyone but you in this thread are taking are the standard "positions" about vector spaces in both math and physics, your apparent expectation that we should "move" from them is mistaken. Particularly when, as you admit, you are expounding your own personal theory. Get your speculations published in a peer reviewed journal and then you can discuss them here.

 

[PeterDonis]

I said that I would not insist on these concepts that mentors are clearly rejecting and so this is over!

Since the "concepts" you refer to are your personal theory, yes, it is indeed over. Thread closed.