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

[Saw]

TL;DR Summary

Is there a plausible argument that can justify treating c as a dimensionless constant?

I once read (though I don’t remember where) that in the same way that you talk about a dimensionless ratio between Y and X in ordinary space, you can conceive of c as a dimensionless ratio between T and X in spacetime.

Do you know where I can find a reliable treatment of that idea?

As clarification:

I am not referring to the possibility that, by choice of units, you can make c = 1. I am aware that simply by choosing seconds as units of time and light-seconds (distance traveled by light in 1 second) or years and light-years, etc, the numerical value of c will be 1, but then it is still a ratio of different units (space units / time units). The idea that I am referring to went further and established that c would be a ratio between the same units, so it would become dimensionless.

I have also seen in A first course in general relativity by Bernhard Schutz, section 1.3, that he proposes to measure time with meters, this meter meaning "the time it takes light to travel one meter". He goes on to say that "if we consistently measure time in meters, then c is not merely 1, it is also dimensionless!". But I am not sure that this is the case. If the meter has that definition, it is still a unit of time. It seems that Schutz's is just another version of the latter idea, where he has also merely ensured that the numerical values of the numerator and the denominator coincide, only through another route: instead of choosing the distance that light travels in 1 second, he has chosen the time that light needs to travel 1 meter, but we are still left with different units for numerator and denominator...

PS1: Maybe after all what Schutz had in mind, even if he did not express it so categorically, is doing two things: (i) simply playing with T (instead of cT) and X, as well as v as a fraction of 1 instead of v/c, just like you do when you choose units where c=1, but without intention of ever coming back to disclose that T and X have different units and (ii) instead of inventing new units for T and X, choosing units of space for both. If so, this reference would be a good peer-reviewed support for the idea that I am referring to and then my questions would be:
- I suppose that the right thing to do is to opt as unit-unificator for either space or time, since otherwise you would create inconsistency with the rest of physics, right?
- Schutz prefers space. Is there a reason why space would be better than time as unit-unificator?

PS2: BTW, section 1.6 of the same book contains a derivation/proof of the invariance of the ST interval without first going through the LTs

 

[Orodruin]

This is solely dependent on your chosen units and the underlying physical dimensions. In SI units, the physical dimensions of length and time are separate. In natural units they are not — length and time have the same physical dimension. Any unit of length is therefore also a unit of time and vice versa. Natural units have many useful properties, such as making the geometric nature of spacetime more evident.

This does not mean that you cannot have both meters and seconds defined in natural units. You can. They are both just units of the same physical dimension, much like meters and feet or seconds and hours. The quantity c is then a unit conversion factor, unity. Much like 3600 s/hr = 1.

Edit: Minor typo fixed.

 

[Dale]

TL;DR Summary: Is there a plausible argument that can justify treating c as a dimensionless constant?

I am not referring to the possibility that, by choice of units, you can make c = 1. I am aware that simply by choosing seconds as units of time and light-seconds (distance traveled by light in 1 second) or years and light-years, etc, the numerical value of c will be 1, but then it is still a ratio of different units (space units / time units). The idea that I am referring to went further and established that c would be a ratio between the same units, so it would become dimensionless.

Actually, what you are talking about is indeed simply a choice of units. Your system of units determines both the numerical values of quantities as well as their dimensionality. The difference between a system of units where the speed of light is numerically 1 but has dimensions and where it is numerically 1 and dimensionless is entirely a matter of arbitrary convention.

 

[Saw]

The difference between a system of units where the speed of light is numerically 1 but has dimensions and where it is numerically 1 and dimensionless is entirely a matter of arbitrary convention.

Understood and agreed, but I am glad that the idea of c as dimensionless is accepted as a valid convention since

Natural units have many useful properties, such as making the geometric nature of spacetime more evident.

I have just found here a discussion about the matter and an opinion points out how in the past sailors had some units for vertical distances (fathoms) and others for horizontal distances (nautical miles), I gather that prompted by the fact that they also used different instruments to measure each.

But in so-called spacetime you use virtually the same instrument to measure time and space, just with a different orientation, like in ordinary space you use the same rulers to measure Y and X distances, just with a different orientation... And in the same way that in ordinary space the Y and X axes are both composed of spatial points, in spacetime T and X axes are composed of events. So all invites to have the same units, as indeed a conventional choice, but a more convenient one.

You did not answer, however, my question about creating new "events" or "spacetime" units. Is it obliged that, if you choose same units for time or space, this be one of the old set, either time or space? (There are some elucubrations on the subject in the document that I linked to.)

 

[PeterDonis]

But in so-called spacetime you use virtually the same instrument to measure time and space, just with a different orientation

This is true in SI units, since we define the meter in terms of the second and a fixed value for the speed of light. Natural units are basically the same except that we use a different fixed value, $1$, for the speed of light, and that we define both dimensions to be the same, whereas SI units define time and length to be different dimensions, just with a fixed numerical relationship between their units.

However, there is still a fundamental physical distinction between timelike and spacelike intervals and vectors, which no choice of units can remove.

 

[Saw]

However, there is still a fundamental physical distinction between timelike and spacelike intervals and vectors, which no choice of units can remove.

Sure. The analogy with ordinary space goes until a certain point.

This is true in SI units, since we define the meter in terms of the second and a fixed value for the speed of light. Natural units are basically the same except that we use a different fixed value, $1$, for the speed of light, and that we define both dimensions to be the same, whereas SI units define time and length to be different dimensions, just with a fixed numerical relationship between their units.

The component of conventionality implied in "we define" is agreed upon: we could act otherwise. But would you also agree on the convenience of some conventions, which is triggered by some physical circumstances? Like the facts that:

- When we define the metre in SI units as the distance that light traverses in 1/299,792,458 s, it helps that we best measure distances through the radar convention.
- When we define the second also in SI units as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom, it is also radiation being the agent.
- When in natural units we define both dimensions to be the same, it helps that the radar convention serves also to synch clocks.

 

[PeterDonis]

- When we define the metre in SI units as the distance that light traverses in 1/299,792,458 s, it helps that we best measure distances through the radar convention.

I believe the rationale for defining the meter in SI in terms of a fixed value for the speed of light was something like this.

- When we define the second also in SI units as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom, it is also radiation being the agent.

Not really, the properties of the hyperfine transition in question are a detailed result of the quantum mechanics of atoms, not just light. That particular transition was chosen, as I understand it, because it is particularly easy to measure accurately, being an outer electron in the heaviest stable hydrogen-like atom.

- When in natural units we define both dimensions to be the same, it helps that the radar convention serves also to synch clocks.

I would say the main rationales for natural units in relativity are: (1) it makes the formulas much simpler since there aren't extra factors of $c$ all over the place, and (2) it makes things easier to conceptualize since units along all dimensions of spacetime are the same (so, for example, the worldlines of light rays in spacetime diagrams are 45 degree lines, and relative speeds translate directly into slopes of worldlines).

 

[Saw]

So, recapitulating, we have mentioned convention:

we define

Also computational convenience:

I would say the main rationales for natural units in relativity are: (1) it makes the formulas much simpler since there aren't extra factors of $c$ all over the place, and (2) it makes things easier to conceptualize since units along all dimensions of spacetime are the same (so, for example, the worldlines of light rays in spacetime diagrams are 45 degree lines, and relative speeds translate directly into slopes of worldlines).

I would like to highlight the physical *constraints* that drive this process of units-unification, in both the substantive sense (the underlying physical reality) and the operational sense (the available measurement technology), thus sometimes forbidding the unification and other times demanding it.

Take this example: imagine that distances are being measured by two groups of people, one who is using miles (for vertical dimension = Y) and another who uses kms (for the horizontal dimension = X).

First realization is that in both cases we are talking about the same reality (distance), even if it is measured from a different angle.

Second is that the two rulers are in essence the same instrument and can be measured against one another and so a conversion rule (named c) can be found. For example, you put a 1-meter ruler one after another over a 1-mile ruler and you see that there are about 1609 meters in a mile, so the conversion factor from miles into km is c = 1.609 km/miles. So when you combine both units through the Pythagorean Theorem, you always have to write c * y miles, like here:

\begin{array}{l}<br /> {s^2} = {c^2}{\rm{ km/miles * }}{y^2}{\rm{ miles}} + {x^2}{\rm{ kms}}\\<br /> {\rm{where c = }}1.609<br /> \end{array}

Third is that the measurements of c progressively improve, but you end up thinking that it is more practical to fix the ratio once and for all and thus define the miles as 1.609344 times a km.

Fourth is that it is more convenient to choose units so that c = 1, e.g. fix as units of X, instead of 1km, a 1km-mile, which is 1.609344 km, so that the formula looks now like this:

\begin{array}{l}<br /> {s^2} = {c^2}{\rm{ km - miles/miles * }}{y^2}{\rm{ miles}} + {x^2}{\rm{ km - miles}} = {y^2} + {x^2}\\<br /> {\rm{where c = }}1<br /> \end{array}

Fifth realization is the final simplification where you decide to either measure in miles or km both X and Y!

In spacetime the “same reality”, being the analog of “distance”, would be “events” and the “same instrument”, playing the part of the “meter”, would be the “light”. (I admit that this picture is rough and may need fine-tuning.)

[Side note: AFAIK, any conversion needs a common reference on which to hinge, no matter if it is a conversion between reference frames or between the axes of the same frame. Can we say that, just like the lightlike vector is the eigenvector, i.e. the axis of rotation, in a transformation, it is also the element on which the conversion btw space and time inside a frame hinges?]

In conclusion, the physical constraints have these implications:

• This process would not take place if length and time were not measuring the same thing, even if from a different angle.
• It would not be possible if you were not measuring with the same instrument, even if oriented in a different manner.
• When these circumstances concur, it is almost a must to complete this process.

Of course, still, as noted, the difference between T and X versus Y and X is that the former combine with a negative sign.

Would you agree that this description of the issue is basically right or want to correct/refine it?

 

[PeterDonis]

Would you agree that this description of the issue is basically right

No. A timelike interval is not the same as a spacelike interval "oriented in a different manner". They are fundamentally different in a way that horizontal and vertical spacelike intervals are not. Because of that, your proposed analogy between the two cases fails.

 

[Ibix]

That seems over-complicated.

The idea that distance and time aren't unrelated things comes from the realisation that an interval one frame calls "a displacement in just time" another frame calls "a displacement in time and space". At that point you can ask if there is a natural conversion factor between units of time and space (obviously a velocity) and $c$ presents itself from the interval equation or the Lorentz transforms.

 

[PeroK]

So, recapitulating, we have mentioned convention:

Also computational convenience:

Then there's geometry.

 

[Saw]

Then there's geometry.

Yes, of course. Geometry is fed with data gathered from measurement instruments that try to capture aspects of reality. So if we start with, as I said, a single reality ("events") and we design our instruments so that they capture aspects thereof (space, time) through an adequate distribution of roles (ideally, with full specialization, so that one performs a job that is perfectly independent of the other), then we can paint those results in a manner that can be manipulated with that wonderful technique that geometry is (for example, by taking advantage that time and space are orthogonal to each other).

But there will not exist orthogonality on the picture if the instruments are not designed to that effect or are not apt for that. Likewise, what we are doing with space and time probably can be done, to a surprising extent, with other units, but there are limits to that. I suppose that there are also a good number of units that are irreconcilable and would not fit into a common coordinate system and hence would not benefit from geometry.

 

[PeroK]

Yes, of course. Geometry is fed with data gathered from measurement instruments that try to capture aspects of reality. So if we start with, as I said, a single reality ("events") and we design our instruments so that they capture aspects thereof (space, time) through an adequate distribution of roles (ideally, with full specialization, so that one performs a job that is perfectly independent of the other), then we can paint those results in a manner that can be manipulated with that wonderful technique that geometry is (for example, by taking advantage that time and space are orthogonal to each other). But there will not exist orthogonality on the picture if the instruments are not designed to that effect or are not apt for that. Likewise, what we are doing with space and time probably can be done, to a surprising extent, with other units, but there are limits to that. I suppose that there are also a good number of units that are irreconcilable and would not fit into a common coordinate system and hence would not benefit from geometry.

That paragraph is totally incomprehensible to me.

 

[Saw]

there is a natural conversion factor between units of time and space (obviously a velocity) and $c$ presents itself from the interval equation or the Lorentz transforms.

A conversion factor that in the end you can dispense with if you measure time and space with the same units, given that you also measure them with the same instrument, albeit oriented in a different manner. All this as Schutz and other authors propose, I am not inventing it...

 

[Saw]

That paragraph is totally incomprehensible to me.

Interesting! What about this: when you draw a coordinate system and do geometry, where do you get the data from? Do they fall from heaven or from what has been measured with a measurement instrument? So if you find that you can manipulate the data with a certain geometry, that is because the instruments work that way and so does the reality that they capture.

Ok, no, I know, you don't need to tell me: this is still garbage for you. Please just keep silent: I will interpret your silence as confirmation.

 

[PeroK]

Interesting! What about this: when you draw a coordinate system and do geometry, where do you get the data from? Do they fall from heaven or from what has been measured with a measurement instrument? So if you find that you can manipulate the data with a certain geometry, that is because the instruments work that way and so does the reality that they capture.

That makes a little more sense, but if geometry is part of the mathematical model on which you build theoretical physics, then you don't need or want strange conversion factors between your coordinate axes. And a ratio of two line segments is dimensionless.

 

[Saw]

No. A timelike interval is not the same as a spacelike interval "oriented in a different manner". They are fundamentally different in a way that horizontal and vertical spacelike intervals are not. Because of that, your proposed analogy between the two cases fails.

Well, analogy is not identity. Are you familiar with how analogies work? It is an exciting subject. Obviously, you don't need that everything works the same in the areas under comparison. There can exist differences. It is enough that the analogy works for certain purposes. For example, it seems obvious to me that just like spatial X and Y, spacetime T and X are axes looking from their respective and orthogonal axes at "events": one of them locally, the other simultaneously. How else do you regard orthogonality btw space and time? If this is not valid for you, can you provide your own vision of perpendicularity in spacetime?

 

[PeroK]

Are you familiar with how analogies work?

Analogies are a poor substitute for mathematics. I would always prefer an isomorphism to an analogy.

PS "spacetime geometry" is not an analogy: it's a precise mathematical model.

 

[Dale]

what we are doing with space and time probably can be done, to a surprising extent, with other units, but there are limits to that

The geometrized units system takes the concept as far as I have seen it taken:

https://en.m.wikipedia.org/wiki/Geometrized_unit_system

 

[Saw]

That makes a little more sense, but if geometry is part of the mathematical model on which you build theorectical physics, then you don't need or want strange conversion factors between your coordinate axes. And a ratio of two line segments is dimensionless.

Well, that is precisely the thing that I am posing since the OP: if in the end, we use the same units for time and space, as some authors propose and looks sound, then c becomes dimensionless and what happens to it?

 

[PeroK]

Well, that is precisely the thing that I am posing since the OP: if in the end, we use the same units for time and space, as some authors propose and looks sound, then c becomes dimensionless and what happens to it?

It vanishes from the mathematical model! It becomes an artifact of the specific units we chose before we knew about GR. In the same way that if we move exclusively to SI units, then the conversion factor of $1.609 \ km$ per mile vanishes from study of distances.

 

[Saw]

The geometrized units system takes the concept as far as I have seen it taken:

https://en.m.wikipedia.org/wiki/Geometrized_unit_system

The table is impressive, it takes the concept quite far!

 

[PeroK]

PS now there's an analogy:

$c = 3 \times 10^8 \ m/s$ is analgous to $1.609 \ km$ per mile. Both conversion factors are required because we've ended up, for historical and/or practical reasons, with two different units where only one was theoretically needed.

 

[PeterDonis]

analogy is not identity.

It's also not valid reasoning. Analogies can sometimes suggest things to look into, but they don't prove anything by themselves.

 

[Saw]

Analogies are a poor substitute for mathematics. I would always prefer an isomorphism to an analogy.

PS "spacetime geometry" is not an analogy: it's a precise mathematical model.

Fair enough, of course, whenever an intuitive likeness or analogy can be turned into a precise mathematical or geometrical concept, it becomes knowledge. But often one thing precedes the other. For example, that is what I tried to do when I suggested that a conversion factor may be a rudimentary form of an eigenvector. This may be an invalid attempt, a stupid thing, but if we managed to link the two things, that would consolidate the idea.

PS now there's an analogy:

$c = 3 \times 10^8 \ m/s$ is analgous to $1.609 \ km$ per mile. Both conversion factors are required because we've ended up, for historical and/or practical reasons, with two different units where only one was theoretically needed.

Please don't tell me that I said something that is not wrong. I had already got used to always getting pushbacks.

 

[PeterDonis]

orthogonality btw space and time?

There is no such thing. Orthogonality is a property of vectors, or worldlines meeting at an event (which comes to the same thing). It is not a property of "space" or "time". It is perfectly possible to find timelike vectors and spacelike vectors that are not orthogonal.

This is what happens when you try to reason by analogy instead of actually looking at the math.

 

[Saw]

There is no such thing. Orthogonality is a property of vectors, or worldlines meeting at an event (which comes to the same thing). It is not a property of "space" or "time". It is perfectly possible to find timelike vectors and spacelike vectors that are not orthogonal.

This is what happens when you try to reason by analogy instead of actually looking at the math.

That is a good point for you all to clarify to me. I have no intention to talk about this in a loose manner, but would like to use the precise mathematical concepts. Aren't the basis vectors of the cT axis (or rather T, on the basis of what is here commented) and the X axis mutually orthogonal, in the so-called Minkowskian or hyperbolic sense, which is that the dot product containing the minus sign between such vectors is 0. If so, that is what I meant. If not, please give me a better mathematical description.

 

[Nugatory]

Aren't the basis vectors of the cT axis (or rather T, on the basis of what is here commented) and the X axis mutually orthogonal,

Only if we choose our coordinates so that the axes are orthogonal.

the so-called Minkowskian or hyperbolic sense, which is that the dot product containing the minus sign between such vectors is 0.

Yes, that’s the definition of orthogonality. That definition is independent of our choice of coordinates - two vectors are orthogonal or not no matter what coordinates we use when we carry out the calculation. But there’s no reason that vectors parallel to a given coordinate axis must be be orthogonal to vectors parallel to some other coordinate axis unless we’ve chosen axes that have that property.

 

[Nugatory]

in the so-called Minkowskian or hyperbolic sense, which is that the dot product containing the minus sign between such vectors is 0.

But please be aware that a better way of describing the dot product is using the metric tensor: the dot product of vectors $U$ and $V$ is $g_{\mu\nu} U^{\mu} V^{\nu}$, where all quantities are expressed in whatever coordinate system we’ve chosen and we’re summing over the repeated indices.

With ordinary Minkowski coordinates $g_{tt}=-1$, $g_{xx}= g_{yy}= g_{zz}=1$, the other twelve components of $g$ are zero and we recover the standard expression for the dot product.

 

[PeterDonis]

Ok, no, I know, you don't need to tell me: this is still garbage for you. Please just keep silent: I will interpret your silence as confirmation.

This is not a constructive attitude. If you keep it up, you will get a warning and your thread will be closed.

 

[Saw]

This is not a constructive attitude. If you keep it up, you will get a warning and your thread will be closed.

Come on, it was a little private joke with Perok, who has been after that most helpful and kind to me. Maybe it was more constructive when you answered my question "can you provide your own vision of perpendicularity in spacetime?" with a "why would I do such stupid thing?", even if you deleted that post, probably when you read my post 27 and realized that I had a basic grasp over the meaning of orthogonality in spacetime? Yes, I did manage to read that post, before you deleted it. There was a time when Physics Forums was a place where people asked questions and others replied, without this tension that your attitude creates. I have been getting very constructive comments from Perok, Dale and now Nugatory, whose post is a perfect orientation on the subject of orthogonality in spacetime that you have refused to provide. Please allow me to keep receiving valuable advice from the people who are willing to give it in PF, who are many. If you don't feel like that, just please kindly leave the thread, but don't threaten me. I don't like that and, especially, I don't need to stand it.

 

[Saw]

Only if we choose our coordinates so that the axes are orthogonal.

Thanks a lot, and tell me please, because this is quite relevant for the object of the thread: would "choosing other coordinates where the axes are not orthogonal" equate to "adopting a different method for measuring space and time"? I mean, different from basically the radar convention for synching clocks and determining distances.

 

[Saw]

But there’s no reason that vectors parallel to a given coordinate axis must be be orthogonal to vectors parallel to some other coordinate axis unless we’ve chosen axes that have that property.

That is most interesting. How can you make that choice?

 

[Nugatory]

Thanks a lot, and tell me please, because this is quite relevant for the object of the thread: would "choosing other coordinates where the axes are not orthogonal" equate to "adopting a different method for measuring space and time"?

How can you make that choice?

We choose whatever coordinates are convenient for the problem at hand. Everyone’s favorite example is the choice between using polar $(r,\theta)$ coordinates and Cartesian $(x,y)$ coordinates when working with the two-dimensional Euclidean surface of a sheet paper. (It would be a good exercise to derive the components of the metric tensor in polar coordinates).

But clearly this choice has nothing to do with the actual distances between points on the sheet of paper or how we measure them - we use a ruler. So to the extent that your first question is well-defined the answer is “No”.

You may have noticed that in polar coordinates $\hat{r}$ and $\hat{\theta}$ are still orthogonal. It is unfortunate that our two most familiar coordinate systems do have orthogonal axes, because we are tempted into the mistaken assumption that orthogonality is a natural property of all coordinate axes. For a counterexample, we need something less familiar: for example, if we’re considering the experience of an observer free-falling into a black hole, the most coordinate system will put the radial zero point at the infaller’s position; in these coordinates the $r$ and $t$ axes are not orthogonal.

 

[PeterDonis]

it was a little private joke with Perok

This explanation is helpful. However, I would still remind you that such things are much harder to get across on a forum than in person, since nobody can see nonverbal things like your facial expression and body language. You can make up for that, to an extent, by using emojis; for example, a smiley face or a wink emoji after the sentences I quoted from you would have made your intent much clearer.

who has been after that most helpful and kind to me.

I have been getting very constructive comments from Perok, Dale and now Nugatory, whose post is a perfect orientation on the subject of orthogonality in spacetime

These acknowledgements are also helpful.

don't threaten me

Drawing your attention to the rules and norms of these forums is not a threat. It's part of my job.

 

[Saw]

We choose whatever coordinates are convenient for the problem at hand. Everyone’s favorite example is the choice between using polar $(r,\theta)$ coordinates and Cartesian $(x,y)$ coordinates when working with the two-dimensional Euclidean surface of a sheet paper. (It would be a good exercise to derive the components of the metric tensor in polar coordinates).

But clearly this choice has nothing to do with the actual distances between points on the sheet of paper or how we measure them - we use a ruler. So to the extent that your first question is well-defined the answer is “No”.

You may have noticed that in polar coordinates $\hat{r}$ and $\hat{\theta}$ are still orthogonal. It is unfortunate that our two most familiar coordinate systems do have orthogonal axes, because we are tempted into the mistaken assumption that orthogonality is a natural property of all coordinate axes. For a counterexample, we need something less familiar: for example, if we’re considering the experience of an observer free-falling into a black hole, the most coordinate system will put the radial zero point at the infaller’s position; in these coordinates the $r$ and $t$ axes are not orthogonal.

Although you answered "no", I am interpreting your answer as "yes". When I equated choosing another system of coordinates with another method for measuring space and time, I did not refer to how we measure the distances between points on the sheet of paper, but to how we obtain (operationally) the data with which we feed the values that we later reflect on the sheet of paper, i.e. which measurements instruments we choose and how we display them.

Even if the decision to analyze a problem from the perspective of another frame is one that we make (as you say, out of convenience, because that makes the answer easier to see) at the desk, actually what you then do is simulate what an observer would obtain after an operational change, which may be more or less dramatic depending on whether the choice involves changing the orientation of the sticks (but keeping them orthogonal) or grabbing sticks that are not orthogonal or grabbing a different instrument like a theodolite or whatever...

The example that you mention illustrates this: taking the infaller's position is an operational change, I would say a most dramatic one.

Now that the question is better defined, would you be able to mention an example of a shift to a non-orthogonal basis (in the context of SR) and how this relates to a change in the nature or the rules of use of the clocks and rulers?

BTW, please let me remind (myself) of the reason for this excursus on orthogonality, for the purpose of retaking "the thread of the thread" in due time: the question was precisely that, in my opinion, it is our progressive understanding of how space and time are built and how they relate to each other at operational level what prompts us to use same units for both dimensions and thus make c dimensionless; in this context, assuming that our operational practice makes them actually orthogonal (even if it could be otherwise), I thought it appropriate to elaborate on the meaning of this orthogonality, because it is something analogous to what happens in ordinary space, where we also normally display X and Y perpendicularly (even if we could do otherwise).

It would also help to know an example or somehow an elaboration on the phenomenon mentioned by PeterDonis: a timelike vector that is parallel to the T axis and is not orthogonal to a spacelike vector that is parallel to the X axis. This would enable me to better understand how orthogonality differs in M-spacetime from orthogonality in Euclidean space, although, in my opinion, it would not ruin the analogy for the particular purpose for which it was conceived.

 

[PeterDonis]

he phenomenon mentioned by PeterDonis: a timelike vector that is parallel to the T axis and is not orthogonal to a spacelike vector that is parallel to the X axis.

That's not what I said. I said you can find timelike vectors and spacelike vectors that are not orthogonal. I did not say those vectors would be parallel to the T and X axes of an inertial frame. That is impossible since those axes are everywhere orthogonal in an inertial frame.

But consider, for example, these two vectors, with $(T, X)$ components given in an inertial frame: $V_1 = (1, 0.1)$, $V_2 = (0.2, 1)$. $V_1$ is timelike and $V_2$ is spacelike, as you can see by computing their squared norms, but they are not orthogonal, as you can see by computing their inner product.

 

[PeterDonis]

how orthogonality differs in M-spacetime from orthogonality in Euclidean space

The general definition of "orthogonal" in vector spaces is that two vectors are orthogonal if their inner product is zero. Minkowski spacetime and Euclidean space are both vector spaces, so they both use that general definition of orthogonality, but their inner products are different (since, being metric spaces, their inner products are derived from their metrics, which are different).

We also sometimes talk about curves being orthogonal (such as the T and X axes of an inertial frame); what we really mean is that their tangent vectors are orthogonal at the point where they meet.

 

[Saw]

That's not what I said. I said you can find timelike vectors and spacelike vectors that are not orthogonal. I did not say those vectors would be parallel to the T and X axes of an inertial frame. That is impossible since those axes are everywhere orthogonal in an inertial frame.

But consider, for example, these two vectors, with $(T, X)$ components given in an inertial frame: $V_1 = (1, 0.1)$, $V_2 = (0.2, 1)$. $V_1$ is timelike and $V_2$ is spacelike, as you can see by computing their squared norms, but they are not orthogonal, as you can see by computing their inner product.

Sorry, I was too tired yesterday to search for your actual statement and misquoted you! No surprise that I was not finding in the internet any examples of what you had not said!

But then this is not so different from what happens in ordinary space, mutatis mutandis, of course, i.e. bearing in mind that the metric and hence the inner product of each vector space is different, as you noted in your next post and I do take into account. (BTW, I sometimes hear that the inner product of Minkowski space is called "bilinear form". Is that a generalized concept of the "inner product", just like the latter generalizes the good old "dot or scalar product"?).

At least, I don't see how this fact should ruin the analogy that I was making between ordinary vector space and Minkowski vector space, to illustrate that our evolution towards using the same units for time and space, and in particular length units for both, as proposed by Schutz and others (geometrical units), is driven by the fact that we are in face of two aspects (space and time) of the same physical reality ("events") and two orientations of the same measurement instrument (perpendicular in the most convenient display, although it could be otherwise, as long as it is not colinear).

At this stage, I don't know what to do. The subject of this excursus is in my opinion relevant to the topic of units-unification. I have ideas about it and would need to test them so as to either reject them or convert the intuition and apparent handwaving into mathematical/geometrical statements. But it is a complicated task in itself, should we discuss it here or should I maybe open another independent thread, about how the concept of orthogonality applies in both realms? I would think it would make sense to continue here, because this gives the discussion on orthogonality a practical background: supporting (or not) the claim in favor of units-unification.

PS: you may say that the matter is solved by noting that the metric in each space and hence the form of its inner product is different; but I think it is needed to go deeper into the reasons and implications of that difference, precisely because otherwise analogy between the two spaces (and its powerful teaching value) is lost; otherwise, noting any difference between the two areas is an obstacle ruining the comparison, whereas the trick with analogies is that there are differences that are relevant for the purpose at hand and others that are irrelevant.

 

[Saw]

I am realizing that I may have left you without much to answer to, so I will be more specific.

I said long ago, speaking about the process of units-unification in SR:

• This process would not take place if length and time were not measuring the same thing, even if from a different angle.
• It would not be possible if you were not measuring with the same instrument, even if oriented in a different manner.
• (...)

Of course, still, as noted, the difference between T and X versus Y and X is that the former combine with a negative sign.

just like spatial X and Y, spacetime T and X are axes looking from their respective and orthogonal axes at "events": one of them locally, the other simultaneously.

And, yes, I do think that here the meaning of orthogonality is the same as you find in ordinary vector space. In the latter, the essential requirement for a basis is that its basis vectors are "linearly independent", meaning that they are not overlapping or colinear; a convenient condition, since it simplifies calculations, is that such basis vectors are orthogonal, that is to say, not only to some extent independent (non-colinear) but "totally independent", meaning that one has 0 components in the others.

We should not abandon this generalized meaning in Minkowski space, just make the necessary adaptations.

The adjustment here is that the points are "events" and consequently both the T axis and the X are bunches of events, although each axis takes care of a specialized or independent function: the T axis is a bunch of events happening locally, at a fixed position x = 0, while the X axis is a bunch of events happening simultaneously, at t = 0; other parallel grid lines do the same thing, at the relevant fixed x or t points, respectively.

You may say then: but orthogonality is invariant, all frames agree on it and in SR all frames would not agree on what you have just stated; instead they do agree on another thing, which is the Minkowskian concept of orthogonality as checked through the Minkowskian dot product. I see no solution to this conundrum other than admitting that there are two versions or concepts of orthogonality: each frame builds its coordinate system assuming that it has orthogonality in the first sense; others disagree with that; but all agree that they all have orthogonality in the second sense, which is fine, because the second one is after all what solves the practical problem under consideration, which is one about causality btw events.

 

[Nugatory]

You may say then: but orthogonality is invariant, all frames agree on it and in SR all frames would not agree on what you have just stated, but they do agree on another thing, which is the Minkowskian concept of orthogonality as checked through the Minkowskian dot product. I see no solution to this conundrum other than admitting that there are two versions or concepts of orthogonality

This “conundrum” will be based on some misunderstanding, but I’m genuinely not sure what your misunderstanding is. There is one concept of orthogonality, applicable everywhere to Euclidean spaces, Minkowski spaces, the more complicated spacetimes of general relativity, everywhere: $g_{\mu\nu} U^{\mu} V^{\nu}=0$. There’s no separate “Minkowskian concept of orthogonality”.

Maybe if you could state this “Minkowski concept of orthogonality”, this “another thing” in the first quoted sentence above? Then we might better understand what you’re thinking?

I do note that you mention frames above as well. I’m not sure what you’re getting at there. Different frames naturally use different coordinates, but using different coordinates does not imply different frames. The choice of units for measurements of time and for distance (which as far as I know is still what this thread is about) is purely a coordinate issue.

Edit to add: The Minkowski spacetime and Minkowski coordinates are different things. People often just say "Minkowski" and rely on the context to clarify which was intended.

 

[malawi_glenn]

Perhaps I am too late to the party here, but you do this in many branches of physics - like in quantum mechanics you set $h=1$ (or $\hbar = 1$). In the end, you know what units you want, so you just insert back as many $h$ (or $\hbar$) as needed to make the units work out.

 

[Saw]

The choice of units for measurements of time and for distance (which as far as I know is still what this thread is about)

Well, here there is confusion because we are mixing the main issue of the thread (units) with the collateral issue of orthogonality. That is why I said in post 39 that maybe we should stop talking about orthogonality here and open a new thread to discuss it. The only link btw the two subjects is that I said that you will be more ready to accept the same units for time and space if you admit that they are two (orthogonal) aspects of the same thing, in the first sense, which I will refer to later as the "intuitive" one.

There is one concept of orthogonality, applicable everywhere to Euclidean spaces, Minkowski spaces, the more complicated spacetimes of general relativity, everywhere: $g_{\mu\nu} U^{\mu} V^{\nu}=0$. There’s no separate “Minkowskian concept of orthogonality”.

Here, in order to build an abstract generalized concept of orthogonality, you are doing it like this: if the inner product of two vectors is 0, then they are orthogonal; a different thing is that, depending on the metric of each space, such dot product may take a different form or (technically) signature. Is this right?

We can call this (conventionally, for lack of a better name) the "Dot Product" concept of orthogonality. This can take a Euclidean or a Minkowskian form, but these (among others) are variants of the same thing.

Then there is another concept, which (again for lack of a better name) I will call the "Independence" sense, which is the one that I described here:

the same as you find in ordinary vector space. In the latter, the essential requirement for a basis is that its basis vectors are "linearly independent", meaning that they are not overlapping or colinear; a convenient condition, since it simplifies calculations, is that such basis vectors are orthogonal, that is to say, not only to some extent independent (non-colinear) but "totally independent", meaning that one has 0 components in the others.

What do you make of this meaning?

On the one hand, this meaning is present in SR, since it is clear to me what I stated here:

points are "events" and consequently both the T axis and the X are bunches of events, although each axis takes care of a specialized or independent function: the T axis is a bunch of events happening locally, at a fixed position x = 0, while the X axis is a bunch of events happening simultaneously, at t = 0; other parallel grid lines do the same thing, at the relevant fixed x or t points, respectively.

On the other hand, I find it hard to accommodate it within the "DP" meaning. It seems to me that it fits with the Euclidean signature but not with the Minkowskian one.

This is the "conundrum": has the "Independence" meaning been dropped behind as a Euclidean thing? If you tell me so, I could accept that this is the case, but then I do believe that each ST reference frame considers itself orthogonal in this "Independence" or "Euclidean" sense, while the others disagree and in turn attribute this feature to themselves. This is simply like when I say that I measure from x = 0, while your origin is located at x = d (in a translation) or when I say that my axes are not rotated, while yours are rotated by angle theta (in a 2D rotation) and vice versa.

 

[PeterDonis]

I sometimes hear that the inner product of Minkowski space is called "bilinear form".

The term "bilinear form" just means any thingie that takes two vectors and spits out a number, and is linear in both of its arguments. The inner product is an example of a bilinear form, but not the only possible one.

this is not so different from what happens in ordinary space

In terms of some pairs of vectors being orthogonal and others not, no, that's a general thing that will happen in any vector space that has an inner product defined on it.

As far as orthogonality goes, the key thing that distinguishes Minkowski spacetime from Euclidean space is that there are vectors--the null vectors--that are orthogonal to themselves.

two orientations of the same measurement instrument

But this does not apply to timelike and spacelike vectors in Minkowski spacetime; they are not just two different orientations of the same measuring instrument. You can't take a ruler and "point it in a timelike direction" to measure time; and you can't take a clock and "point it in a spacelike direction" to measure distance. Because of the way SI units are defined, you can use light in both your clock and your ruler (or more precisely to calibrate both your clock and your ruler) if you use those units, but that still doesn't make them the same measuring instrument.

 

[PeterDonis]

Here, in order to build an abstract generalized concept of orthogonality, you are doing it like this: if the inner product of two vectors is 0, then they are orthogonal; a different thing is that, depending on the metric of each space, such dot product may take a different form or (technically) signature. Is this right?

We can call this (conventionally, for lack of a better name) the "Dot Product" concept of orthogonality. This can take a Euclidean or a Minkowskian form, but these (among others) are variants of the same thing.

Then there is another concept, which (again for lack of a better name) I will call the "Independence" sense

No. These are not two different concepts of orthogonality. There is just one concept of orthogonality, the "Dot Product" concept, which is perfectly well defined: two vectors are orthogonal if their dot product is zero. This is well-defined regardless of whether the dot product can have negative values or not (in Minkowski spacetime it can, in Euclidean space it can't).

The other concept that you are calling "independence" is the standard concept of linear independence, not orthogonality: two vectors are linearly independent if neither one is a scalar multiple of the other. More generally, a set of vectors is linearly independent if none of them can be expressed as a linear combination of the others. This is a necessary property for a set of vectors to be a basis of the vector space (the other necessary property is that the set must span the space, i.e., there cannot be any other vector in the space that is linearly independent of the set).

These are two different concepts and there is no confusion between them. If you are not familiar with them, I would suggest taking some time to study vector spaces and their properties.

It is often convenient to choose a basis for a vector space in which all of the vectors are orthogonal to each other (and it is also often convenient to have all of them be unit vectors, which is what the term "orthonormal basis" refers to--the vectors are all orthogonal to each other and all normalized to be unit vectors). But this is not required for a basis; it's just often convenient. The only required properties for a basis are the ones I gave above: that the basis vectors are all linearly independent and that they span the space.

At this stage, I don't know what to do.

I would suggest making sure you are thoroughly familiar with vector spaces and their properties, especially properties like orthogonality and linear independence, before you try to apply these concepts to the subject you seem to be interested in, which is systems of units.

I would not suggest trying to handwave definitions or applications for these vector space concepts on your own. This is a thoroughly studied subject and you should know the standard concepts in it and their standard definitions and applications.

 

[PeterDonis]

Different frames naturally use different coordinates, but using different coordinates does not imply different frames.

This depends on what definition you are using for "frame"--one possible definition is "coordinate chart", in which case different coordinates would imply different frames.

The definition you are implicitly using here for "frame" is "frame field", i.e., a continuous mapping of orthonormal tetrads to events. Then you could change coordinates without changing frames--you could match up a different coordinate chart to the same set of tetrads, by changing the units of at least one of the coordinates. In Minkowski spacetime there isn't much reason to do this except for pedagogy, but things get more complicated in curved spacetimes.

 

[Saw]

I would suggest making sure you are thoroughly familiar with vector spaces and their properties, especially properties like orthogonality and linear independence, before you try to apply these concepts to the subject you seem to be interested in, which is systems of units.

I would not suggest trying to handwave definitions or applications for these vector space concepts on your own. This is a thoroughly studied subject and you should know the standard concepts in it and their standard definitions and applications.

Well, nothing of what you have mentioned about vector spaces is something that I have not studied and indeed deeply studied. To convince me that I should drop my point and retire to study vector spaces, you should point out a specific point in which I am mistaken. But, without pretending that I know all about the subject, I am walking on quite firm ground in the area to which the discussion is restricted.

For example, these things that you are explaining to me, I have already brought them up myself, with other words:

two vectors are linearly independent if neither one is a scalar multiple of the other. More generally, a set of vectors is linearly independent if none of them can be expressed as a linear combination of the others. This is a necessary property for a set of vectors to be a basis of the vector space (the other necessary property is that the set must span the space, i.e., there cannot be any other vector in the space that is linearly independent of the set).

It is often convenient to choose a basis for a vector space in which all of the vectors are orthogonal to each other (and it is also often convenient to have all of them be unit vectors, which is what the term "orthonormal basis" refers to--the vectors are all orthogonal to each other and all normalized to be unit vectors). But this is not required for a basis; it's just often convenient. The only required properties for a basis are the ones I gave above: that the basis vectors are all linearly independent and that they span the space.

What I find difficult to understand is why you don't recognize that there is a link between so-called linear independence (the essential thing, as I said) and orthogonality (the convenient thing, as I also said) and this link is one of degree:

- if two vectors are only linearly independent, but not orthogonal, it means that they are not colinear or occupying the same line, but if you project one over the other, you still find that one projects a shadow over the other, i.e. it has some component of the other (that is why their dot product is not zero);
- instead, if two vectors are orthogonal, it means the same thing to a higher extent: if you project one over the other, you find that one does not project any shadow over the other, i.e. it has no component of the other (that is why their dot product is zero).

Do you see why I say that it is a question of degrees: some component vs. no component, some shadow vs no shadow?

If we cannot agree on this elementary thing, then I will concur (constructively quoting your own words) that one of us must go and make sure that he/she is thoroughly familiar with vector spaces and their properties, especially properties like orthogonality and linear independence, before continuing with the discussion.

 

[PeroK]

- if two vectors are only linearly independent, but not orthogonal, it means that they are not colinear or occupying the same line, but if you project one over the other, you still find that one projects a shadow over the other, i.e. it has some component of the other (that is why their dot product is not zero);
- instead, if two vectors are orthogonal, it means the same thing to a higher extent: if you project one over the other, you find that one does not project any shadow over the other, i.e. it has no component of the other (that is why their dor product is zero).

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.

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.

 

[PeterDonis]

To convince me that I should drop my point and retire to study vector spaces, you should point out a specific point in which I am mistaken.

Sure, here's an example:

What I find difficult to understand is why you don't recognize that there is a link between so-called linear independence (the essential thing, as I said) and orthogonality (the convenient thing, as I also said) and this link is one of degree:

Because there is no such "link". The two concepts are different concepts. @PeroK's comments in this regard are good ones.

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 we cannot agree on this elementary thing, then I will concur (constructively quoting your own words) that one of us must go and make sure that he/she is thoroughly familiar with vector spaces and their properties, especially properties like orthogonality and linear independence, before continuing with the discussion.

Yes, and that person would be you. See above.

 

[Saw]

Dears, I have to leave for a while now, not to study vector spaces, but to the Gym and later dinner. I will reply on return. Hope that the thread is not closed by then!