Abstract or Physical?

[FrumiousBandersnatch]

Then I'll wait for the issue to be settled. However, in the meantime, I'd wager that if spacetime is declared to be equivalent to quantum foam, a supporting abstraction will develop to explain why that is - there will be yet another turtle standing beneath and supporting the current one.

If you don't get the reference, I'm suggesting an infinite regress.

It's generally acknowledged that whatever the fundamental building blocks of reality are/is, it's not a coherent question to ask what they're/it's made of. AIUI the current consensus is that it's probably quantum fields. Some things just have to be brute facts.

 

[quaternion]

It's generally acknowledged that whatever the fundamental building blocks of reality are/is, it's not a coherent question to ask what they're/it's made of. AIUI the current consensus is that it's probably quantum fields. Some things just have to be brute facts.

Maybe I just ask the question in an odd, unscientific manner and you're clairvoyant enough to get my meaning, but I get the impression not all on this forum accept that position.

 

[FrumiousBandersnatch]

Maybe I just ask the question in an odd, unscientific manner and you're clairvoyant enough to get my meaning, but I get the impression not all on this forum accept that position.

I don't really see how it could be otherwise - unless, as you suggested, there is an infinite regress of scale and corresponding emergence... but even if that's the case, there will still be some limit of accessibility and description for practical purposes ¯\_(ツ)_/¯

 

[sjastro]

QFT is generically known as relativistic quantum mechanics and is the combination of SR (not GR) with QM and has led to highly accurate theories such as QED (Quantum Electrodynamics).
The reason why SR is used is that curved spacetimes in GR are based on a Lorentz manifold which is another way of stating at local or small scales curved spacetime looks like flat spacetime; much like the Earth is spherical but appears flat at small scales which can be described by Euclidean geometry.
QED is based on scales of typically no greater than the Compton wavelength of an electron which is around 2.43 x 10⁻¹² m.
At this scale curved spacetime can be considered flat.

As I mentioned in a previous post at this scale where QED applies spacetime can exert a pressure via the Casimir effect which is quite remarkable for a mathematical abstraction.
Furthermore this has nothing to with Quantum foam which is at the Planck scale of 1.6 x 10⁻³⁵m where spacetime behaves quantum mechanically.
Once again this is remarkable for a mathematical abstraction.
Spacetime in the form of a vacuum is a quantum field in the lowest energy state E yet at the Planck scale it behaves like a particle at or below its Compton wavelength according to the energy-time version of the Heisenberg uncertainty principle ΔEΔt ≥ h/4.

It is at the Planck scale that scientists are theorising “what spacetime is made out of” which is another counter to the idea spacetime is nothing more than a mathematical abstraction.
https://www.scientificamerican.com/article/what-is-spacetime/

 

[quaternion]

I don't really see how it could be otherwise - unless, as you suggested, there is an infinite regress of scale and corresponding emergence... but even if that's the case, there will still be some limit of accessibility and description for practical purposes ¯\_(ツ)_/¯

I considered starting a new thread, but since activity on this one has died down, and not wanting to pepper the forum with all my questions, I decided to extend this one. Just be forewarned I'm somewhat changing directions.

At one point it seemed you at least implied that if spacetime were a thing, that means we should be able to measure its properties. If that's what you meant, I agree. I would go even further, though. Whether spacetime is an abstraction to explain the material behavior of other things, or a thing in and of itself, the implication is that it has properties - either secondarily due to the things it explains or intrinsically.

First, a small digression: currently, units of time (seconds) are defined based on the frequency of cesium-133. If physics reaches the point of identifying spacetime as a thing, would the definition of the second be changed to coincide with that?

Once a thing is understood as having properties, it is possible those properties can change. Even though, as best we know it, the speed of light is constant, the rest mass of all electrons is the same, etc. there is hypothetically a possibility those properties could vary. Recall that the apparent constancy of the properties of the electron, in part, led to the one electron postulate.

There are, of course, advantages to assuming the properties of time are constant - of selecting the most stable of elements as the basis for its definition. There are, however, also some advantages to allowing the properties of time to change based on the material properties of the system studied.

 

[FrumiousBandersnatch]

At one point it seemed you at least implied that if spacetime were a thing, that means we should be able to measure its properties. If that's what you meant, I agree.

I don't think that was me; if you can quote or link to my post, that would help.

I would go even further, though. Whether spacetime is an abstraction to explain the material behavior of other things, or a thing in and of itself, the implication is that it has properties - either secondarily due to the things it explains or intrinsically.

The way I see it, spacetime is primarily a model we have made to explain our observations. This model is successful because it provides an explanatory framework for those observations and makes fruitful predictions about other observations we should expect to make in various circumstances. IOW, given the data we currently have, it is reasonable to say that our spacetime model usefully describes important aspects of what we are observing. For linguistic and conceptual convenience, we equate our model with whatever is causing the phenomena we observe that resulted in our model - IOW we say we are modelling spacetime.

Whether that means spacetime is a 'thing' or not, I don't know. Concepts like 'real' and 'thing' are hard to pin down, and typically have ambiguous definitions that vary with context. The properties of spacetime are, presumably, those phenomena that generate the patterns in the observations we make. If they differ from what our model predicts, we modify the model; if they significantly contradict the model, we may need to make a new model that is a better fit. If we have to make a new model, we'll probably change its name to reflect that.

First, a small digression: currently, units of time (seconds) are defined based on the frequency of cesium-133. If physics reaches the point of identifying spacetime as a thing, would the definition of the second be changed to coincide with that?

Why would we want to change it? why would the 'thingness' of spacetime be relevant?

Once a thing is understood as having properties, it is possible those properties can change. Even though, as best we know it, the speed of light is constant, the rest mass of all electrons is the same, etc. there is hypothetically a possibility those properties could vary. Recall that the apparent constancy of the properties of the electron, in part, led to the one electron postulate.

There are, of course, advantages to assuming the properties of time are constant - of selecting the most stable of elements as the basis for its definition. There are, however, also some advantages to allowing the properties of time to change based on the material properties of the system studied.

I really don't know what you mean. Time is what clocks (e.g. the frequency transitions of caesium 133) measure. Our model of spacetime tells us that time is dynamic rather than absolute, and our observations have confirmed those expectations.

 

[essentialsaltes]

First, a small digression: currently, units of time (seconds) are defined based on the frequency of cesium-133. If physics reaches the point of identifying spacetime as a thing, would the definition of the second be changed to coincide with that?

Units are arbitrary. Milk is a thing. We measure it in gallons. ¯\_(ツ)_/¯

 

[quaternion]

Units are arbitrary. Milk is a thing. We measure it in gallons. ¯\_(ツ)_/¯

Units are not completely arbitrary. Yes, of course we can use gallons or liters to measure volume, but the dimensionality of volume is always distance cubed. We can't just arbitrarily decide volume should be distance squared plus time.

There are some historical examples of people misunderstanding the dimensionality of units and making poor choices. The easiest example is the equivalence of the English pound mass with pound weight - something that gave me no end of headaches in school.

I don't think that was me; if you can quote or link to my post, that would help.

No, that's fine. If I misattributed, my apologies.

Whether that means spacetime is a 'thing' or not, I don't know. Concepts like 'real' and 'thing' are hard to pin down, and typically have ambiguous definitions that vary with context.

True, but it does have implications.

Why would we want to change it? why would the 'thingness' of spacetime be relevant?

I really don't know what you mean. Time is what clocks (e.g. the frequency transitions of caesium 133) measure. Our model of spacetime tells us that time is dynamic rather than absolute, and our observations have confirmed those expectations.

You are correct, so I suppose an example is in order.

When studying the vibration of structures, the typical equation (case 1) uses a linear differential equation with trig functions for a solution. However, this solution doesn't correlate well to all cases. As such, some nonlinear equations (case 2) have been proposed as an alternative. The interesting consequence of nonlinear equations is that they imply the potential for chaos. Sometimes tests demonstrate chaos for these systems, sometimes they don't.

In cases where neither the circular trig functions nor the nonlinear functions seem an appropriate solution, there is yet a third possibility - (case 3) Jacobi elliptic functions. In all 3 cases, the argument of these functions is taken to be time. Yet, with elliptic functions one faces an interesting choice.

The elliptic sine can be denoted sn(t,k), where t = time, k = modulus. If denoted this way, the solution is nonlinear and becomes just a special version of case 2.

The elliptic sine can also be denoted sin(am(z,k)), where, if Non-Newtonian calculus is used, time = am(z,k) and the solution is linear ... no chaos implied, yet a better fit than case 1. The philosophical implication is that time is being redefined to suit the properties of the system (i.e. its eccentricity).

 

[FrumiousBandersnatch]

True, but it does have implications.

What is the 'it' and what are the relevant implications it has?

When studying the vibration of structures, the typical equation (case 1) uses a linear differential equation with trig functions for a solution. However, this solution doesn't correlate well to all cases. As such, some nonlinear equations (case 2) have been proposed as an alternative. The interesting consequence of nonlinear equations is that they imply the potential for chaos. Sometimes tests demonstrate chaos for these systems, sometimes they don't.

In cases where neither the circular trig functions nor the nonlinear functions seem an appropriate solution, there is yet a third possibility - (case 3) Jacobi elliptic functions. In all 3 cases, the argument of these functions is taken to be time. Yet, with elliptic functions one faces an interesting choice.

The elliptic sine can be denoted sn(t,k), where t = time, k = modulus. If denoted this way, the solution is nonlinear and becomes just a special version of case 2.

The elliptic sine can also be denoted sin(am(z,k)), where, if Non-Newtonian calculus is used, time = am(z,k) and the solution is linear ... no chaos implied, yet a better fit than case 1. The philosophical implication is that time is being redefined to suit the properties of the system (i.e. its eccentricity).

Sorry, I can't follow that argument - I don't see how it implies time is being redefined, or what 'redefining' time is supposed to mean.

 

[quaternion]

What is the 'it' and what are the relevant implications it has?

The way time is used in models is one implication, and I'm trying to explain that.

Sorry, I can't follow that argument - I don't see how it implies time is being redefined, or what 'redefining' time is supposed to mean.

I hope you don't mind, then, if I spread this out over several posts - do the step by step thing. If it starts to bog down, though, I don't want to destroy what's been a nice conversation.

Studying motion means studying change over time. If we're going to do that, we have to be able to measure time. All metaphysical arguments aside, the practical implications are that we compare the motion of the test subject to the motion of cesium-133. Yes? Or, even more practically, we compare to a count of ticks from an electronic digital device. In doing so, we're picking a device that we hope makes those ticks at a constant rate.

 

[FrumiousBandersnatch]

Studying motion means studying change over time. If we're going to do that, we have to be able to measure time. All metaphysical arguments aside, the practical implications are that we compare the motion of the test subject to the motion of cesium-133. Yes? Or, even more practically, we compare to a count of ticks from an electronic digital device. In doing so, we're picking a device that we hope makes those ticks at a constant rate.

OK.

 

[quaternion]

OK.

Our model of time, then is t = az, where 'a' is constant and 'z' is some discrete counting of periodic events. We also assume a transform from discrete in z to continuous in t.

The example I gave (mechanical vibration) is also periodic. We know, with the same certainty we have for our model of time, that the measured system has a constant period of P seconds. If we were only interested in results every P seconds, everything would be heavenly. But since our model of time is continuous, we assume our model of the mechanical system is also continuous.

In short, we query the model for results at any arbitrary time. That's where the issue emerges.

 

[FrumiousBandersnatch]

Our model of time, then is t = az, where 'a' is constant and 'z' is some discrete counting of periodic events. We also assume a transform from discrete in z to continuous in t.

The example I gave (mechanical vibration) is also periodic. We know, with the same certainty we have for our model of time, that the measured system has a constant period of P seconds. If we were only interested in results every P seconds, everything would be heavenly. But since our model of time is continuous, we assume our model of the mechanical system is also continuous.

In short, we query the model for results at any arbitrary time. That's where the issue emerges.

So how does that mean redefining time?

 

[sjastro]

So how does that mean redefining time?

Yes I was wondering that myself and given clocks are the issue here, the relevant equation involves Simple Harmonic Motion;
y = sin(nt + ε)
And the period T the number of seconds per cycle is;
T = 2π/n

 

[quaternion]

So how does that mean redefining time?

I'm not there yet. I just wanted to see if you agreed with each of the steps. Assuming that's the case ...

Just because a system is periodic doesn't mean that periodicity can be described with simple trig functions. When simple trig functions don't work, the assumption that often follows is that the system is nonlinear. However, nonlinear solutions, as I mentioned, often introduce chaos. In other words, the model no longer presents a strictly periodic system, but a nearly periodic one. That violates my stipulation that the system is strictly periodic.

We need a solution that is strictly periodic, yet doesn't have a circular orbit in the phase plane (i.e. something other than trig functions).

 

[FrumiousBandersnatch]

I'm not there yet. I just wanted to see if you agreed with each of the steps. Assuming that's the case ...

Just because a system is periodic doesn't mean that periodicity can be described with simple trig functions. When simple trig functions don't work, the assumption that often follows is that the system is nonlinear. However, nonlinear solutions, as I mentioned, often introduce chaos. In other words, the model no longer presents a strictly periodic system, but a nearly periodic one. That violates my stipulation that the system is strictly periodic.

We need a solution that is strictly periodic, yet doesn't have a circular orbit in the phase plane (i.e. something other than trig functions).

OK - although, IIRC, mechanical vibration is only strictly periodic under specific conditions...

 

[hedrick]

I know people will say it matches observations of physical reality, but you can't really "observe" things like spacetime. It's not like you can put a piece of spacetime in a test tube and play with it. You can't say, "Look here's how light behaves over here when it interacts with spacetime, and this is how it behaves over there without spacetime."

You can't put space-time in a test tube, but you can see how light behaves in the vicinity of various massive objects, and test it against the predictions of general relativity. Over the last century or so physicists have been very inventive about finding ways to test general relativity and other models. So far general relativity looks quite good.

The Wikipedia article General relativity - Wikipedia mentions a number of predictions and many of the tests.

 

[quaternion]

You can't put space-time in a test tube, but you can see how light behaves in the vicinity of various massive objects, and test it against the predictions of general relativity. Over the last century or so physicists have been very inventive about finding ways to test general relativity and other models. So far general relativity looks quite good.

The Wikipedia article General relativity - Wikipedia mentions a number of predictions and many of the tests.

I don't disagree.

 

[quaternion]

OK - although, IIRC, mechanical vibration is only strictly periodic under specific conditions...

Yes, though it is typically modeled as such due to the convenience of doing so. Some certification societies require it.

The simplest alternative to circular functions are elliptic functions. However, in order to maintain the linear character - the strict periodicity - they must be written in a form like: x = sin(am(z,k)).

The 'z' refers to the discrete counting I mentioned before, and so our time model is now: t = am(z,k), which is different. We have redefined time in terms of the elliptic character of the system. This time model does not progress at a constant linear rate with respect to z as did the first model.

https://www.mathworks.com/help/symbolic/jacobiam_plot.png

 

[sjastro]

Yes, though it is typically modeled as such due to the convenience of doing so. Some certification societies require it.

The simplest alternative to circular functions are elliptic functions. However, in order to maintain the linear character - the strict periodicity - they must be written in a form like: x = sin(am(z,k)).

The 'z' refers to the discrete counting I mentioned before, and so our time model is now: t = am(z,k), which is different. We have redefined time in terms of the elliptic character of the system. This time model does not progress at a constant linear rate with respect to z as did the first model.

https://www.mathworks.com/help/symbolic/jacobiam_plot.png

You have not redefined time at all.

The application of elliptic functions has its origins in the physics behind pendulums.
As a first order approximation the mathematical formula describing the period of a simple pendulum is.

The formula is reasonably accurate for small amplitudes where the angle of release θ₀ is small.

However by taking into consideration the kinetic plus potential energy of a swinging pendulum is constant everywhere along its path of travel one can derive an equation
(well beyond the scope of this post) the following;

Here F is an elliptic function of the first kind defined as;

The advantage with this formula it applies to any arbitrary amplitude and θ₀ can be large.

As an example if the pendulum has a length l=1 metre and is released at an angle of 10⁰
and g= 9.8 m/s².
The first order approximation is T₀ ≈ 2.0064 s, whereas as T ≈ 2.0102 s.
The differences have nothing to do with having to redefine time, T gives a more accurate value than T₀.