Classical Dynamics: Appreciating Reference Frames
A desperate attempt at making reference frames interesting
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. — Pierre Simon Laplace, Essays on Probability
Before we begin, I must confess that this article primarily serves as an interlude; we’ll avoid most of the mathematics and instead, just focus on what reference frames really mean and why they’re important. But to appreciate them in all their glory, we must first recount some of their history.
The Good Stuff
The fundamental principles of classical mechanics were laid down by Galileo and Newton in the 16th and 17th centuries. In writing Principia, Newton gave us the three laws of motion and in writing Dialogue on the Two Chief Systems of the World, Galileo offered us a deeper insight into the nature of celestial bodies — probably the single greatest scientific achievement in history, you might think. You might also think this pretty much wraps it up for classical mechanics. And, in a sense, it does. Newton and Galileo, along with others (shout out Lagrange), created an unyielding map that answered all that Man was then concerned with: the motion of planets, stars, and of galaxies.
Through the course of time, classical mechanics has been as useful as it has been reliable. Given a collection of particles, acted upon by a collection of potentials, you draw a nice little diagram with arrows that show the forces acting on the bodies. You then add up your forces and let F = ma figure out the future positions and momenta of your system of particles. All you need is enough patience and a big enough computer and you’re done. With the turn of the 19th Century, however, things changed. More than anything, people came along asking “If we say something about our physical world, does our statement remain true in a different reference frame?”. In other words, “Is an observation on the ground the same as one made from the sky? What quantities change? What quantities don’t?”. Consequently, the answers to these questions sparked a revolution in the early years of the 20th Century. As a result, special relativity emerged and physics triumphed once again. But what even are reference frames and why are they so important?
Reference Frames
You must, at some level, already be familiar with reference frames. Cartesian coordinates, for example, are familiar to most: a Cartesian frame consists of a set of spatial coordinates x, y, and z. If you wish to imagine coordinate frames concretely, you may think of space as a lattice of rulers, such that you can define each point by a combination of lengths. This is a spatial coordinate system, allowing us to define where an event occurs. This is a little unsatisfactory to the modern scientist since it is often important to equally consider when an event is observed. To this end, you also need a time coordinate. A reference frame is then a coordinate system allowing room for both space and time, consisting of the x, y, z, and t axes. If you wish to, you can also extend your notion of reference frames by imagining each point in space with a clock; these clocks are synchronized at t = 0 and all run at the same rate, therefore allowing a measurement of when an event has been observed. But of course, there are many ways to specify points in space and time, allowing many different reference frames.
To begin with, one could consider translating the origin to some other point. So, instead of x = y = z = t = 0 being the origin, one could equally choose any other point and measure all events relative to it. You can also rotate coordinates about some point to yield a different orientation. And finally, you can consider frames moving relative to one another. But this isn’t important right now; we’ll get into it later. What’s worth noting is that we can speak of your frame and my frame and besides the axes and origin, each frame can be different.
This is perhaps better illustrated with an example. We’ll consider the usual train adage. Imagine Alice on the train platform and Bob on the moving train. Each of them has their own lattice of rulers and clocks that they carry with them. Alice is at the center of her reference frame and Bob is at the center of his. Obviously, the coordinates are different. Alice specifies the coordinates of an event by some x, y, z, and t; Bob specifies the same event by a different set of coordinates that account for the fact that Bob’s reference frame is moving with respect to Alice’s. If, for example, Bob’s train moves in the y-axis, Alice and Bob will not agree on the y-axis coordinate. Bob would always attest that his nose ends at y = 3, meaning that it is 3 feet away from the center of his head. Alice would disagree. Alice would say that the position of Bob’s nose is changing with time. Bob could also scratch his head at some time, t = 4. And you might think that Alice’s clock would read the same but that’s where relativistic physics departs from Newtonian physics. The assumption, although intuitive, that all clocks in all frames of reference can be synchronized is a false one. For now, however, let’s assume that Newton’s physics holds and all clocks are indeed synchronized with each other even if they have relative motion.
This brings us to inertial frames of reference.
Inertial Frames
You may have heard before that inertial reference frames are those in which Newton’s laws hold. Ludwig Lange, the man who first coined the very term “inertial reference frame”, defines it as the following:
A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.
But what does that mean anyway? When the dust settles, I encourage you to adopt the following understanding: an inertial reference frame is one that does not undergo any acceleration. By extension, it is a reference frame in which a particle experiencing no net undergoes motion with a perceived constant velocity. For those that prefer even further concreteness, a free particle with constant mass in an inertial reference frame travels in a straight line defined by r = a + vt where the three-vectors are bolded: r defines the position, a is the initial position, v is the velocity vector and t, the time. Notice how for zero velocity v = 0, an object at rest will remain at rest, giving us back Newton’s first law of motion. It then follows that any reference frame moving with a velocity relative to an inertial reference frame is also an inertial reference frame. Let me run through that again. For two frames of reference moving at constant velocity with respect to each other, there would be no acceleration between the two. You can convince yourself that this is true by considering a number of cases. In fact, the first postulate of special relativity is precisely this. Einstein writes in The Foundation of the General Theory of Relativity, Section A
If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K’ moving in uniform translation relatively to K
Measurements in one inertial frame can so be converted to measurements in another by a simple transformation, the Galilean transformation in classical physics and the Lorentz transformation in relativistic physics. The reason why Galilean transformations do not work in relativistic physics is perhaps beyond the scope of this article but linked here is a deeper and more complete assessment of reference frames. Intuitively, however, I hope you understand that reference frames are those in which the laws of physics are the most simple. In fact, the principle of relativity postulates that the laws of physics are invariant of their reference frames. This is often attributed to Galileo for his work on Galilean relativity.
Galilean Transformations
Galilean transformations are used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group. Below is a brief introduction to them.
An inertial reference frame is not unique in itself. Consider S an inertial reference frame. There are ten transformations such that S → S’ is also an inertial reference frame.
But the three spatial transformations are not unique in themselves. But, for most purposes, they are the only interesting ones. The others are transformations of form x’ = λx for some λ ∈ R. This is just a rescaling of the coordinates. For example, we may choose to measure distances in S in units of inches and distances in S′ in units of meters. Combined with the principle of relativity, each transformation is telling us something important about the Universe.
Translations tell us that there is no special point in the Universe. Rotations tell us there is no privileged direction. And boosts tell us there is no special velocity. Physics is the same regardless (within classical limits, of course). The first two are fairly unsurprising: position is relative; direction is relative. You define your position and direction with respect to other bodies. In itself, it means little to say I live four kilometers. You must specify from where and in which direction. The “no special velocity” rule tells us that there is no such thing as absolute stationariness. You can only be stationary with respect to something else. But the transformations don’t end here.
Let’s unpack things. First, orthonormal matrix? What does that mean? In linear algebra, an orthonormal matrix is a real square matrix whose columns and rows are orthonormal vectors (vectors perpendicular to one other). In other words, equally, a matrix is orthonormal if multiplying it by its own transpose yields the identity matrix. This is just fancy talk to convey the fact that orthonormal matrices rotate a reference frame. Below are three examples of matrices that rotate a coordinate grid by an angle, θ. The first represents rotation about the x-axis, the second represents rotation about the y-axis, and the third is rotation about the z.
Below is an animation showing the respective rotations and how a reference frame remains an inertial frame under rotational transformations.
This brings us to translations. A Galilean translation is an equivalent of moving a coordinate grid by some units in any of the three dimensions. It is obvious to see how the reference frame remains inertial even under these transformations. A velocity boost then follows as a continuous translation. Below is an animation depicting both transformations since they are essentially synonymous. But it is important to note that a Galilean boost changes your reference frame. A translation only changes your position in the same reference frame.
We’ve missed out on the time transformation. The time translation, compared to the spatial transformations, is a little different. Galilean invariance of time refers to the concept that time intervals between events are the same in different inertial frames of reference. In simpler terms, the measurement of time remains consistent across different frames of reference that are moving at constant velocities, as long as the velocities are non-relativistic. If observers in two inertial frames, S and S′, fix the units — seconds, minutes, hours — in which to measure the duration time then the only remaining choice they can make is when to start the clock. In other words, the time variable in S and S′ differ only by a constant, which is the difference between when they start the clock. The time interval between the two events, however, will remain just the same. This suggests that the fundamental laws of physics do not care about when you begin your clock; they will act the same regardless. All evidence suggests that the laws of physics are the same today as they will be tomorrow.
With that, I will end this here. As always, thank you for reading!
