In trying to understand and resolve the paradox I wrote about in the first post in this series, I will be taking a somewhat circuitous route in order to lay some important groundwork before we can directly confront the paradox.
We can start the journey by looking at one of the most fundamental concepts in physics, that of the nature of space. On the surface, space seems like a very straightforward concept. It is seen as a kind of container in which everything in the universe exists. But difficulties arise when one asks questions such as whether space can be viewed as something positive, a tangible entity that has its own properties that can be detected, or whether it is viewed as something negative, that signifies the absence of matter in a region. Another way of posing the distinction is asking whether, if one can conceive of removing all the matter and energy in the universe, what would we be left with? Just ’empty space’? In the absence of matter, would such a thing have any meaning at all?
To address this question, we can start by examining the nature of the motion of matter through space. That brings us to the issue of what are called inertial frames that are intimately connected with the nature of space. Many of the laws of physics are believed to be valid only in inertial frames but it is not easy to identify those frames and part of the difficult lies in the fact that the nature of space is hard to pin down.
A ‘frame’ is short for ‘reference frame’ and refers to any system an observer uses to measure the motion of objects. The surface of the Earth is often used as a reference frame because it is the one we live in and are most comfortable with but you could just as easily use a moving car or train or boat or plane and measure the motion of any object with respect to it. When we get to relativity, it is necessary to be more specific about what a frame consists of and it is usually described as a system of standardized rulers and clocks that are spread out over all of space and move with the observer. An ‘event’ is any occurrence that all observers will agree about (such as my picking up a cup or a snap of my fingers) and is uniquely specified by the ruler and clock readings at the location of that event, since only one event can occur at any place at any time.
Different observers that are in motion with respect to one another will have their own sets of rulers and clocks that move with them. A key point is that while all observers will see the same events, we cannot assume that the location and time of any given event as measured by the rulers and clocks of one observer will have the same values when measured by the rulers and clocks of another observer who is moving relative to the first. So while events are unique, the space-time parameters that identify them will be different, their differences depending on whether the two frames are moving with low relative constant speeds (in which case we can use Galilean transformations), or at constant speeds that are appreciable when compared to the speed of light (in which case we must use Lorentz transformations), or they have a relative acceleration (in which case we must use general relativity). Those distinctions will not concern us in this series of essays until very much later.
Most people are familiar with some formulation of the law of inertia, also known as Newton’s first law of motion, that says that an object will continue in its state of rest or constant speed in a straight line unless acted upon by a net external force. (The technical term ‘velocity’ is used denote the combination of both speed and direction.) An inertial frame is one is which that law is seen to hold. (One can read more about the law of inertia at https://www.britannica.com/science/law-of-inertia)
But identifying which frame is inertial and which is not is tricky because a subtle circular reasoning can enter, leading to a sort of chicken-and-egg problem. An inertial frame can be identified as one in which the law of inertia is seen to hold for an object. But the law of inertia only holds in an inertial frame. In other words, we can say that we are in an inertial frame if we observe that an object that is subjected to a zero net external force is at rest or moving with constant velocity. But the problem is how would we know that there is no net force on the body? We know this if it is at rest or moves with constant velocity. But that is true only if we are in an inertial frame. That is the circular logic. This subtlety is usually elided when teaching physics, especially at the introductory level. (This is justifiable since deep subtleties can be mystifying and off-putting to novice learners, but we need to come to grips with it in this series of posts.)
Can one eliminate the circularity? Isaac Newton tried to do so by saying that space is something fixed and an inertial frame was a frame that was at rest with respect to that space or moving with a uniform velocity through it. Newton’s idea of space can be used to try and identify an inertial frame intuitively. I recall one of my professors in college saying that in an inertial frame you ‘feel at home’ and can ‘drink a cup of tea’ with no problems. (This was in Sri Lanka where tea metaphors are common.) That homespun image was meant to evoke the fact that a person in an inertial frame can tell if they are accelerating or not. If you are in a car that is at rest or traveling at constant speed in a straight line (i.e., with constant velocity), you can drink a cup of tea without spilling. But if the car changes its speed by braking or accelerating or changing direction (i.e., if it changes its velocity), then the tea sloshes around and can spill. We can know that the car’s velocity is changing purely by what happens within the car without even looking out the window. Changes in velocity always indicate accelerations and this means that the frame is not inertial. Thus we can feel whether we are in an inertial frame or not. So when we are at rest on the Earth or moving with constant velocity horizontally with respect to the Earth’s surface, we can feel that we are in an inertial frame.
But that is not strictly true because the surface of the Earth is not an inertial frame. The Earth is spinning on its axis and is also orbiting the Sun, and rotational motion always introduces accelerations. One can add another layer to the rotational motion by noting that our Solar system lies in a spiral arm of the Milky Way galaxy and that arm too is rotating about its center. And then we can also add the motion of the galaxy as a whole. The reason we do not sense these accelerations is because they are small, too small to detect in our everyday lives. The acceleration due to the spinning of the Earth about its axis is 0.034 m/s2 and that due to its orbital motion about the Sun is 1.2×10-5 m/s2 which are small compared to everyday accelerations like that due to gravity on the surface of the Earth, which is 9.8 m/s2. Such small accelerations can only be detected with sensitive measuring instruments. If you have a frame that is at rest or traveling horizontally on the surface of the Earth with constant velocity, then for all practical intents and purposes, that acts approximately as an inertial frame since the effects of the Earth’s spinning about its own axis and its orbital motions around the Sun are so small.
The smallness of those non-inertial effects is both fortunate and unfortunate. It is fortunate because we believe that Newton’s laws of motion and the laws of electromagnetism codified in Maxwell’s equations are strictly valid only in inertial frames. They could be (and were) discovered and formulated in the frame of the Earth, even though it is not an inertial frame, because the non-inertial effects are so small. If the non-inertial effects had been significant, then the simple forms that these laws take in inertial frames would have been much harder to discern. On the other hand, it is unfortunate because it likely prevented us from getting to grips with the true character of inertial frames for a long time.
The small effects due the Earth’s spinning can be detected using careful measurements (Newton did experiments to demonstrate them as I will discuss later) and they also show up in large scale systems like the atmosphere where they affect air flows, an effect that we commonly ascribe to ‘Coriolis forces’. This is one of the so-called ‘fictitious forces’ or ‘inertial forces’ that are introduced to explain the deviations from the behavior expected in inertial frames using Newton’s laws and ‘real’ forces, i.e., forces whose sources we can identify such as attached ropes and springs and gravity. The fictitious Coriolis force is due to the Earth’s rotation that makes it a non-inertial frame and it causes air currents to appear to veer away from straight line motion. Another fictitious force is the so-called ‘centrifugal force’ that is used to explain why, when we are in a car and take a sharp turn, we feel as if we are being ‘pulled’ outwards. For the people inside the car, it appears as if a mysterious force has suddenly kicked in.
It is the presence of these small fictitious forces that alerts us that we who are stationary on the surface of the Earth, despite appearances to the contrary, are not in an inertial frame.
(To be continued.)