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.

  • 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.

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!

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.

“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

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https://www.grc.nasa.gov/WWW/K-12/airplane/newton.html
  1. 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.”
  2. 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.”
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All forces accounted for, balanced and not accelerating

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.

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.

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Space-Time Coordinates with infinite extents. Space holds a 3:1 dimensional ratio to time.

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.

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3 measurements at 3 different times allow us to unwind a spherical vector into 3D cartesian space

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.

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.

  • 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.
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If mass is a derived unit, it cannot fit into an abstract frame with infinite limits.
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In a derived frame, time is already embedded with all our spatial dimensions.

Is Time for a Better Reference Frame?

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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.

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We can recycle “time” and use it with “change” to add more spatial resolution to our frames of reference

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!

Expert Modeler | Scientist | Teacher | Engineer

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