Constant v Invariant

[studiot]

There have been several threads recently where some dubious statements about the distinction between constant and invariant have been made.

I am therefore posting this thread to examine this issue by discussion.

I am kicking off with a question

 

An observer is watching the rear end of a train which is receeding a high speed , but not accelerating.

Hanging on the back of the train by a long coil spring is a lamp which is oscillating up and down.

What is the relativistic effect on the spring constant, ie what is the difference, if any, between the spring constant according to the train guard and the observer ?

[Markus Hanke]

1 hour ago, studiot said:

What is the relativistic effect on the spring constant, ie what is the difference, if any, between the spring constant according to the train guard and the observer ?

It changes by a factor \(\gamma^{-1}\), due to transformation of 3-forces perpendicular to direction of motion.

[studiot]

So it is not constant in all cases ?

Nor is it an invariant of the system.

[swansont]

1 hour ago, studiot said:

So it is not constant in all cases ?

Differing between the train and the observer is an issue of invariance. Being a constant was not addressed.

[studiot]

1 hour ago, swansont said:

Differing between the train and the observer is an issue of invariance. Being a constant was not addressed.

I'm sorry I don't follow.

As the lamp is osscillating up and down its travel distance lies at right angles to its motion as part of the train so it is unaffected by relativity.

But as part of the train its receeding motion affects the observer's perception of the period of the oscillation in accordance with relativity.

So its period will appear to increase (as Markus says).

But its period depends on the spring constant

Quote

Mass on a spring - Where a mass m attached to a spring with spring constant k, will oscillate with a period (T). Described by: T = 2π√(m/k).

 

University of Birmingham.

Since the period increases k must decrease.

So it is a different spring constant.

[swansont]

2 hours ago, studiot said:

I'm sorry I don't follow.

The answer referred to observers in different frames. That’s an issue of invariance. Being constant was not mentioned. Being invariant does not imply that something is constant (mass, for example)

[StringJunky]

1 hour ago, swansont said:

The answer referred to observers in different frames. That’s an issue of invariance. Being constant was not mentioned. Being invariant does not imply that something is constant (mass, for example)

Invariance is when one calculates, say a transformation, and the observed parameter (constant) remains unchanged. Is that correct?

[swansont]

57 minutes ago, StringJunky said:

Invariance is when one calculates, say a transformation, and the observed parameter (constant) remains unchanged. Is that correct?

Yes; the transform is between frames of reference (which is what Markus described).

The parameter need not be a constant, and constant can refer to to a universal constant (won’t change over time) or something that’s merely unchanging under a specific set of circumstances.

[StringJunky]

4 minutes ago, swansont said:

Yes; the transform is between frames of reference (which is what Markus described).

The parameter need not be a constant, and constant can refer to to a universal constant (won’t change over time) or something that’s merely unchanging under a specific set of circumstances.

Thanks. Context matters.

[studiot]

5 hours ago, swansont said:

The answer referred to observers in different frames. That’s an issue of invariance. Being constant was not mentioned. Being invariant does not imply that something is constant (mass, for example)

I didn't say it was and I agree that one of the main uses of the term invariant is the comparison of something between different coordinate systems or frames.

I tried to avoid some of the more esoteric uses and offer a solid physical one.

The physics of tha lamp  oscillation is form invariant between the frames.

But that form include a coefficient which is constant, but that constant has a different value is different in every frame, so the constant itself is not invariant.

 

I was also trying not to point the finger at individuals but since you are commenting on Markus, I think he was to enthusiastic when he wrote this.

On 11/6/2023 at 7:40 AM, Markus Hanke said:

Constancy means that c always has the same value under all circumstances - which it evidently does not, since its value depends on the permittivity and permeability of the underlying medium. For example, c is different in glass than in vacuum. This is a direct result of Maxwell’s equations.

all circumstance ?

No any more than the spring constant has the same value under all circunstances.

It too is constant under a ( more limited) specific set of circumnstances in that it is constant over a range of extensions of the spring.

 

Anyway if you don't like that example try analysing the oft quoted 'constant AC voltage'.

Or perhaps the example I gave before, which has nothing to do with coordinate systems or frames.

My freezer keeps a constant temperature, but this temperature is not invariant as I am at liberty to turn the temperature maintained up or down.

Or a final one

Most of physics can be described by differential equations.

We obtain 'solutions' to these DEs by integrating them.

But integration includes an arbitrary constant of intgration, which is not therefore invariant.

Of course we can eliminate the constant if we work by difference as in the definite integral or in the case of relativity coordinate differences.

 

These words have a lot of work to do.

 

 

Edited November 7, 2023 by studiot

[Killtech]

46 minutes ago, studiot said:

all circumstance ?

No any more than the spring constant has the same value under all circunstances.

It too is constant under a ( more limited) specific set of circumnstances in that it is constant over a range of extensions of the spring.

 

Anyway if you don't like that example try analysing the oft quoted 'constant AC voltage'.

Or perhaps the example I gave before, which has nothing to do with coordinate systems or frames.

My freezer keeps a constant temperature, but this temperature is not invariant as I am at liberty to turn the temperature maintained up or down.

Or a final one

Most of physics can be described by differential equations.

We obtain 'solutions' to these DEs by integrating them.

But integration includes an arbitrary constant of intgration, which is not therefore invariant.

Of course we can eliminate the constant if we work by difference as in the definite integral or in the case of relativity coordinate differences.

 

These words have a lot of work to do.

the constantly or isotropy of the speed of light always refers to the constant \(c\) as it appears in the Maxwell equations for vacuum and it is strictly constant by postulate. the speed of light in a medium is not - and most text name it as such so not to mistake it with the (vacuum) speed of light.

Due to its specific context to the vacuum Maxwell equations, the constancy or isotropy of it in all frames almost uniquely fixes the form of that equation in every frame to the same shape (you could argue that hypothetically the ratio between \(\epsilon_0\) and \(\mu_0\) could change in the equations... but no), which practically implies it has to be invariant. In reverse, the invariance assures that all constants that appear in the equation stay the same. So for constancy of \(c\) and invariance of vacuum Maxwell are almost equivalent in SR.

\(c\) is different from a spring constant which definition isn't that strict, such that it may change with rising temperatures.

More generally in order to be able to speak about constancy, a quantity must be represented as a value or some other mathematical structure for which such a relation is even defined. objects in reality aren't made out of number and it is only measurement that associates them with numbers. Not all measurement methods are per se guaranteed to be consistent to each other, i.e. distances can be measured by in units of a rod or the time light in vacuum takes to travel that distance, or they could be just given as a difference of coordinate - one may find two distances to have same length with one method but mismatch with another. 

physical frameworks usually define everything clearly enough, including the valid methods of measurement leaving no ambiguity and therefore within such a framework it is clear if a physical entity is a constant or not. what defines such a framework is not just the laws of physics, but also a lot of technical definitions and conventions. Therefore, it may be a question of representation rather then physics if a quantity is constant or not. 

[studiot]

10 hours ago, Killtech said:

the constantly or isotropy of the speed of light always refers to the constant c as it appears in the Maxwell equations for vacuum and it is strictly constant by postulate. the speed of light in a medium is not - and most text name it as such so not to mistake it with the (vacuum) speed of light.

Due to its specific context to the vacuum Maxwell equations, the constancy or isotropy of it in all frames almost uniquely fixes the form of that equation in every frame to the same shape (you could argue that hypothetically the ratio between ϵ0 and μ0 could change in the equations... but no), which practically implies it has to be invariant. In reverse, the invariance assures that all constants that appear in the equation stay the same. So for constancy of c and invariance of vacuum Maxwell are almost equivalent in SR.

c is different from a spring constant which definition isn't that strict, such that it may change with rising temperatures.

More generally in order to be able to speak about constancy, a quantity must be represented as a value or some other mathematical structure for which such a relation is even defined. objects in reality aren't made out of number and it is only measurement that associates them with numbers. Not all measurement methods are per se guaranteed to be consistent to each other, i.e. distances can be measured by in units of a rod or the time light in vacuum takes to travel that distance, or they could be just given as a difference of coordinate - one may find two distances to have same length with one method but mismatch with another. 

physical frameworks usually define everything clearly enough, including the valid methods of measurement leaving no ambiguity and therefore within such a framework it is clear if a physical entity is a constant or not. what defines such a framework is not just the laws of physics, but also a lot of technical definitions and conventions. Therefore, it may be a question of representation rather then physics if a quantity is constant or not. 

 

Basically I agree with pretty well all you have said here.

I would just stress that there are many types of invariants (see my answer to StringJunky below)  invariants are not functions, there is no' invariant function', as there is a 'constant function'  in maths.

Invariants are a property of certain types of functions or transformations.

 

16 hours ago, StringJunky said:

Invariance is when one calculates, say a transformation, and the observed parameter (constant) remains unchanged. Is that correct?

Yes

Invariants are a property of certain types of functions or transformations.

As such they are the foundation of the modern way that relativity is viewed.

But not all invariants work the same way.

For instance the fractal invariant or the scale invariant is an invariant of geometry that does not related to a coordinate frame like relativity.

There is an interesting modern view of fractal invariants in Neural Nets and the Brain from Oxford University (in pdf) that I am about to post in that long runningargument over artificial consciousness.

https://academic.oup.com/cercor/article/33/8/4574/6713293

 

[Meilos]

The spring constant, k, in Hooke's law (F = -kx) is a measure of the stiffness of the spring. It's important to note that the spring constant is a property of the material and geometry of the spring itself, not a result of the forces applied to it.

[Bufofrog]

6 hours ago, Meilos said:

The spring constant, k, in Hooke's law (F = -kx) is a measure of the stiffness of the spring. It's important to note that the spring constant is a property of the material and geometry of the spring itself, not a result of the forces applied to it.

How does that apply to the discussion?

[studiot]

7 hours ago, Meilos said:

The spring constant, k, in Hooke's law (F = -kx) is a measure of the stiffness of the spring. It's important to note that the spring constant is a property of the material and geometry of the spring itself, not a result of the forces applied to it.

Note that since T = 2π√(m/k). the period also depends upon the mass of the lamp.