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:

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

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:

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

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.