Part 1: Schrodinger’s Next Lesson… Fix the Broken Reference Frame
Introduction
Just a few weeks ago I dived into a research project about Schrodinger and uncertainty. I discovered that “Schrodinger’s Cat” was a thought experiment, devised by Schrodinger, to point out what was conceptually wrong with the accepted interpretation of the double-slit experiment. I looked around but never pinned down what the official error was so I decided to move forward with my research on uncertainty. So, I started with Newton and his 3 laws of motion and kept my eye out for anything suspicious.
It didn’t take very long before my plan unraveled along with the known universe. I’ve been losing sleep from both excitement and stress trying to find the seam or seams that are leaking precious Space-Time into the aether. It’s been a roller coaster of emotions but from what I can tell the leaky seam is in and around our definition of a proper frame of reference and its leaking time. Of course, the leak has been around a while and has trickled into other areas making a mess that history has helped obscure. This is what I have so far:
- Our definition of an inertial reference frame needs a more precise definition.
- Assumed simultaneity in measurements is likely contributing to the “uniqueness” problem.
- Our concept of time is too dependent on space for it to act like a truly independent variable and would benefit from redefinition.
While I don’t know if these are THE problems between classical and quantum mechanics, there are interesting questions and ideas here that would be beneficial to address and answer along with a compilation of some of my research and notes to help avoid starting at the beginning.
This article is Part 1 of 2. This article is largely about frames of reference while Part 2 will focus on the interpretation of the Double-Slit Experiment.
Statistical Mechanics 101 with Dr. Schrodinger
I can’t help but feel like I am in Schrodinger’s posthumous Statistical Mechanics class. It’s ironic that I started this journey with the intent to teach about uncertainty but seem to have lost certainty myself along the way. I wonder what Schrodinger would tell me to do? I think this lesson is for me this round and he’d say something like, “Go figure it out. Oh, and make sure we don’t have to learn this lesson again”. “Roger that”, I’d say!
I don’t really know what I am getting myself into but this problem we have in physics is too important a thing to leave lying around and not get others to take a look at. There is an interesting symmetry between how I have learned to manage and understand uncertainty and how I see others manage it. I see that same symmetry here in physics. I’m curious to see where this goes.
Lastly, I need your help. I’m not a physicist and so I’m traditionally not qualified to weigh in. It’s a big topic so your eyes and ideas are welcome. Who knows, maybe we can discover some new physics together.
Defining a Reference Frame?
The idea of an inertial reference frame took a while to sink in. It was only after going through the motions of trying to recreate a reference frame, that Newton might have had in front of him, that it started to sink in. Comparing my college years to what I’m seeing and doing now has uncovered differences between what I was taught and what Newton might have meant. While the differences have been surprising and informative, there is really no way to pin down what is “right” since reference frames have been poorly understood and thus hotly debated since Newton’s time (1642–1726). It doesn’t help that the idea of an inertial reference frame didn’t come around until after Newton in the late 1800s.
Let’s start with our current definition of a proper Inertial Frame of Reference:
“A ‘frame of reference’ is a standard relative to which motion and rest may be measured… An inertial frame is a reference-frame with a time-scale, relative to which the motion of a body not subject to forces is always rectilinear and uniform...” — SEP
Intuitive Definition
Okay, we are looking for a perspective that will give us the best view of our problem. My understanding is that we want a frame that will hold one or more bodies independently of each other such that motion, rest, and acceleration can all be determined and easily if possible.
What I want to do is rebuild a Newtonian reference frame step-by-step and talk through what I find interesting as well as address any issues. I want to follow what Newton might have done as closely as possible so I can get the best view of Dr. Schrodinger’s homework assignment. Let’s review Newton’s 3 laws of Motion next.
These are Newton’s 3 Laws as we usually see them. However, I was fortunate enough to have stumbled upon a source that had a direct translation of Newton’s original Latin manuscript which is reproduced below. There are noticeable differences that make me wonder how well history has played telephone with physics.
Newton’s 3 Laws Directly Translated:
- “LAW I: An object at rest will remain at rest unless acted upon by an external and unbalanced force. An object in motion will remain in motion unless acted upon by an external and unbalanced force”.
- “LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. — If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.”
- LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. — Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.”
Takeaways From the Original Translation
Law I tells me how objects should be behaving in a proper frame. Unless there is an external force, objects should be at rest or in a steady state of motion. This includes position and or velocity of the translational and or rotational variety.
Law II tells us that force is linearly proportional to velocity (F ∝ v). Interestingly, Law 2 doesn’t mention acceleration or even mass. History is already starting to get foggy. I proved this to myself on paper but I can’t understand or find why this law was interpreted to be Force = mass * acceleration specifically. In trying to prove it to myself I found that a viable alternative would have defined force as Force(alt) = density * velocity (F = 𝞀𝝼) where density, rho, would be the linear operator that modifies velocity proportionally for any object. This is almost identical to our definition of momentum except mass would be a density (momentum = mass*velocity). Interestingly, if force was the product of density and velocity it would look identical to a conservation law where force would be 3 dimensional and conserved. Perhaps someone else can weigh in here?
Law III tells us that forces are equal and opposite. This must be Newton’s strategic choice for developing calculus. Using simple concepts in probability, equal and opposite would account for and establish limits for all the known forces in a given frame. Even if your measurements don’t catch it, your model and experimental evidence would point directly to where you need to look and tell you exactly how much you are missing. This is incredibly brilliant!
Interpretation
I have always seen Newton’s 3 laws taught independently of each other with each having their specific purpose. They are the 3 rules of doing physics the Newtonian way. Going through it again in detail I am not sure Newton intended them to be held separate or independent. For all the controversy surrounding what a proper inertial reference frame is, Newton’s 3 laws seem to make the most sense when held together and also seem to be all ingredients needed to construct a proper inertial reference frame.
Regardless of where all the science lands, I have to say that I am consistently impressed at how well all my favorite scientists did without calculators, computers, and Wiki.
How to Frame A Problem — An Analysis of Frames Reference
Aside from Newton’s 3 laws, a frame of reference needs a coordinate system. I’ll trust our modern definition since Newton’s 3 laws don’t mention it explicitly.
In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame. — Wiki
It has been a huge challenge trying to find a way to see through Newton’s eyes. There are several complications and they are all history related. One complication involving our unit system comes from Joseph Fourier, who gave us all the concept of physical dimensions in 1822. This was done nearly 100 years after Newton’s death. While I am sure some form of units existed, I have no way of knowing how well they were defined and or converted over time. As a result, it has been a headache trying to retrace the steps that got us here. Additionally, Newton’s work, which was published in his “Philosophiæ Naturalis Principia Mathematica” in 1687, is in Latin and doesn’t contain explicit formulae for his 3 laws which were unusual for his time.
Newton’s Space-Time Coordinates
Regardless of your preference of units, our first abstract units and their dimensions are our time and space dimensions. If Newton had celestial bodies in mind then he must have had a Space-Time coordinate system like the one below.
Newton wanted to understand gravity so he needed to add mass somewhere. To do this he’d likely aim to establish mass as a third and independent variable/axis that could be defined relative to time and space.
Just to refresh anyone who needs it, the mathematical definition of a dimension, that is independent, is identical to a free axis or degree of freedom, DF. Thus, an independent variable belongs to an axis perpendicular to any other axis. More simply, any dimension that is independent has a multiplicative relation to any other dimension.
Note on Base Units: I’d imagine that anything that is defined as a fundamental unit/dimension would need to satisfy all the requirements needed to be treated as another perpendicular axis to both time and space. More specifically, a fundamental unit needs independence. This definition doesn’t fit our current set of “base” units but could be an important classification method or way to group units in the future.
A Fundamental Ratio and Symmetry
Newton’s base units and abstract coordinates are time and length. With space being 3D and time linear, or 1D, there is a 3:1 dimensional ratio that governs how we define and map functions that relate one axis to the other. This ratio also tells us how much information to expect in each relation we define. Our Space-Time Coordinate System here dictates that space holds a cubic relation to time. This strikes me as a fascinating rule of symmetry that our mathematics has imposed on our concept of Space-Time and our reality.
The intuitive rule here is that a “unique coordinate” must contain 1 dimension of time for every 3 dimensions of space.
For us, that would mean time must be at least two points (a point has no dimensions whereas a line is 1D) and an elapsed time would map to each of our three space dimensions and points. This makes sense to me but a graphic will help prove it.
If we consider an object in space, such as the Earth, then in the graphic below I imagined taking three different measurements on the surface of the Earth.
If we plan on using calculus to integrate over time we have to assume that time can be broken down into infinitely small time steps. The obvious consequence here is that the chances of any time measurement being simultaneous with another would be zero. You can double-check that by trying to integrate over any probability density function, PDF, at a single point. It will always be zero. This has strong implications for arguing true simultaneity in measurements. Is there some other way to justify it as I am not sure?
The three points we take together give us all the information needed to find a position, velocity (rotational and translational [R&T]), and acceleration (R&T) along with any moments to place your zero wherever you need it to be.
In the Space-Time coordinate system Newton gave us, it appears we want to avoid simultaneity. If it is a system of linear equations where each point carries information about all position related variables (for 3 measurements anyways), entering the same time or forgetting to put zeros in for other vectors, like rotational velocity, would prevent your function from converging to a unique solution.
Since there are quite a few “non-uniqueness” problems for our solutions to 3D differential equations, my favorite being Navier-Stokes, I’d like to see if ensuring there is a unique time to each space coordinate, even if it’s small, is enough to fix the issue. I’m working on checking this myself!
To establish a frame you need to derive position, velocity, and acceleration from the combination of your first 3 points. Doing this is also a check to tell you if your chosen anchor for your frame is of inertial reference quality or not.
Options and Variety: Classifying Frames of Reference
Perhaps you noticed that the reference frame defined above had infinite extents in time and space. Since there aren’t definitive boundaries to either of those two dimensions, we have to anchor our reference frame to some object that obeys our three conditions, or laws, and take measurements from that relative and derived position.
This is precisely what I think Galileo, Newton, and Einstein tried to tackle, along with many others, with their theories of relativity. Any theory of relativity seems to need to address how and why a frame should be anchored so that standardized measurements can/could be taken from them reliably.
Abstract Reference Frame vs Derived Reference Frame
When thinking about how we can try to attach our concept of mass to our Space-Time Reference Frame I discovered a couple of interesting problems and features that will need development in the future to help keep things organized.
So far, this is how I have organized them to keep them straight:
- Abstract Frame — This frame holds the rules and is a theoretical template. It also establishes a conceptual limit and anchor for zooming out of a derived frame to view projections that we may want to integrate over.
- Derived Frame — This frame has limits, that are defined in terms of the abstract frame, and any reductions in dimensionality obey the ratio established by the abstract frame.
Let’s see how this works.
A negative mass doesn’t make sense so it won’t fit nicely into a frame with infinities in it. The only option is to put mass in a finite or derived frame that is a “child” to this more theoretical abstract frame.
A frame with finite limits, where we can place everything in terms of 0–1 or 0–100%, is needed to hold mass. Mass as an independent variable should have a linear relation to time and a cubic relation to space so as not to violate the rules established by our abstract frame. Notice that mass per unit of space gives us exactly density (mass/volume) and per unit of time would be a mass flux. This seems to work out intuitively as well as follow the dimensionality rule in our abstract frame.
This is where I started to encounter trouble with our concept of time. In a finite or derived reference frame, I noticed that time didn’t fit the same way it did in our abstract reference frame. In a derived frame, time would have already been mapped to space, to establish a reference, so I’m not completely clear how to add time again. It would be problematic if we lost track. There appears to be an issue with how time is defined.
It makes me anxious just thinking about it because I don’t know how we would be able to clearly distinguish an embedded time concept in our derived frame from and additional time reference in the same frame. If they got mixed up somewhere our units of “seconds” have no clear way to distinguish which frame it comes from or belongs to.
Unfortunately, after a few days of thinking about this, I can’t seem to tease it all apart from the way it is now. The only way I can think to make sense of it is by redefining our concept of time.
Is Time for a Better Reference Frame?
In terms of relativity, our concept of time doesn’t descritize well inside a reference frame. Time passes the same for all objects giving it a sort of infinite extent. But, time can’t pass differently for objects in a frame giving time no spatial resolution. If you have ever heard that 1 human year is equal to 7 dog years then you can see that we already try to overcome this limitation. On the other hand, space doesn’t share this limitation and has no problem on its own. Thinking about time as some dimension between objects in a frame only makes sense when space dimensions are there to help. This makes me wonder how fundamental our concept of time is and whether or not it should be considered a fundamental unit or if it should sit next to the derived ones.
Time for Change…
When considering a new definition for time I wanted to make sure that our old definition would still work and be useable. It would just be reclassed to a derived unit and have a more specific definition to keep them separate.
To start I asked myself, “What IS time and what does it do?” What I understand is that time is the stick we use to measure and make sense of change. Time, as we understand it now, is a crude measure of many changes that happen all at once and on many scales at different rates. It seems that time is itself at the end of a spectrum which makes it valuable as a potential conceptual limit.
What is “change”? Change is the distance between points on the lifecycle of a thing or an event. Change has the benefit of infinite granularity but can still borrow that infinite extent that comes with time. Looking below we can see that they pair well and help algebraically solve for any missing pieces.
Swapping time and change for each other is something we already do and even in daily conversation. For example, a conversion might look like changing 1 year to 1 rotation around the sun. Change would simply refer to what happened over some period of time.
Much more consideration will certainly be needed to make any change of this magnitude. I am still working on this myself along with a few ideas to improve the rest of our reference frame.
Ending Thoughts
We started with trying to define a proper reference frame and we ended with breaking time. Even after all this, I am not sure how to define an inertial frame of reference with these things in the air. I am still working on it and welcome any advice, correction, and or help!
In Part 2 and up next, Schrodinger gives me more homework involving our interpretation of the double-slit experiment. My job is to present you with enough information to determine whether or not we missed something important and whether or not that important thing leads us to a new theory of gravity.
