Category: SCIENCE

Physics is the study of dynamics. Dynamics is the description of the actual forces of nature that, we believe, underlie the causal structure of the Universe and are responsible for its evolution in time. We are about to embark upon the intensive study of a simple description of nature that introduces the concept of a force, due to Isaac Newton. A force is considered to be the causal agent that produces the effect of acceleration in any massive object, altering its dynamic state of motion.

Newton was not the first person to attempt to describe the underlying nature of causal ity. Many, many others, including my favorite ‘dumb philosopher’, Aristotle, had attempted this. The major difference between Newton’s attempt and previous ones is that Newton did not frame his as a philosophical postulate per se. Instead he formulated it as a mathemat ical theory and proposed a set of laws that (he hoped) precisely described the regularities of motion in nature.

In physics a law is the equivalent of a postulated axiom in mathematics. That is, a physical law is, like an axiom, an assumption about how nature operates that not formally provable by any means, including experience, within the theory. A physical law is thus not considered “correct” – rather we ascribe to it a “degree of belief” based on how well and consistently it describes nature in experiments designed to verify and falsify its correspon dence.

It is important to do both. Again, interested students are are encouraged to look up Karl Popper’s “Falsifiability 29 and the older Postivism30. A hypothesis must successfully withstand the test of repeated, reproducible experiments that both seek to disprove it and to verify that it has predictive value in order to survive and become plausible. And even then, it could be wrong!

If a set of laws survive all the experimental tests we can think up and subject it to,

we consider it likely that it is a good approximation to the true laws of nature; if it passes many tests but then fails others (often failing consistently at some length or time scale) then we may continue to call the postulates laws (applicable within the appropriate milieu) but recognize that they are only approximately true and that they are superceded by some more fundamental laws that are closer (at least) to being the “true laws of nature”.

Newton’s Laws, as it happens, are in this latter category – early postulates of physics that worked remarkably well up to a point (in a certain “classical” regime) and then failed. They are “exact” (for all practical purposes) for massive, large objects moving slowly com pared to the speed of light3³1 for long times such as those we encounter in the everyday world of human experience (as described by Sl scale units). They fail badly (as a basis for prediction) for microscopic phenomena involving short distances, small times and masses, for very strong forces, and for the laboratory description of phenomena occurring at rela tivistic velocities. Nevertheless, even here they survive in a distorted but still recognizable form, and the constructs they introduce to help us study dynamics still survive.

Interestingly, Newton’s laws lead us to second order differential equations, and even quantum mechanics appears to be based on differential equations of second order or less. Third order and higher systems of differential equations seem to have potential problems with temporal causality (where effects always follow, or are at worst simultaneous with, their causes in time); it is part of the genius of Newton’s description that it precisely and sufficiently allows for a full description of causal phenomena, even where the details of that causality turn out to be incorrect.

Incidentally, one of the other interesting features of Newton’s Laws is that Newton in vented calculus to enable him to solve the problems they described. Now you know why calculus is so essential to physics: physics was the original motivation behind the invention of calculus itself. Calculus was also (more or less simultaneously) invented in the more useful and recognizable form that we still use today by other mathematical-philosophers such as Leibnitz, and further developed by many, many people such as Gauss, Poincare, Poisson, Laplace and others.

In the overwhelming majority of cases, especially in the early days, solving one or more problems in the physics that was still being invented was the motivation behind the most significant developments in calculus and differential equa tion theory. This trend continues today, with physics providing an underlying structure and motivation for the development of much of the most advanced mathematics.

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We are now finally prepared to tackle a very difficult concept. All of our dynamics so far is based on the notion that we can formulate it in an inertial frame. It’s right there in Newton’s Laws – valid only in inertial frames, and we can now clearly see that if we are not in such a frame we have to account for pseudoforces before we can solve Newtonian problems in that frame.

This is not a trivial question. The Universe doesn’t come with a frame attached – frames are something we imagine, a part of the conceptual map we are trying to build in our minds in an accurate correspondence with our experience of that Universe. When we look out of our window, the world appears flat so we invent a Cartesian flat Earth. Later, further experience on longer length scales reveals that the world is really a curved, approximately spherical object that is only locally flat -a manifold⁶⁷ in fact.

Similarly we do simple experiments suspending masses from strings, observing blocks sliding down inclined planes, firing simple projectiles and observing their trajec tories – under the assumption that our experiential coordinates associated with the Earth’s surface form an inertial frame, and Newton’s Laws appear to work pretty well in them at first. But then comes the day when we fire a naval cannon at a target twenty kilometers to the north of us and all of our shots consistently miss to the east of it because of that pesky coriolis force – the pseudoforce in the rotating frame of the earth and we learn of our mistake.

We learn to be cautious in our system of beliefs. We are always doing our experiments and making our observations in a sort of “box”, a box of limited range and resolution. We have to accept the fact that any set of coordinates we might choose might or might not be inertial, might or might not be “flat”, that at best they might be locally flat and inertial within the box we can reach so far.

In this latter, highly conservative point of view, how do we determine that the coordi nates we are using are truly inertial? To put it another way, is there a rest frame for the Universe, a Universal inertial frame S from which we can transform to all other frames S’, inertial or not?

The results of the last section provide us with one possible way. If we systematically determine the force laws of nature, Newton tells us that all of those laws involve two ob jects (at least). I cannot be pushed unless I push back on something. In more appropriate language (although not so conceptually profound) all of the force laws are laws of interac tion.

I interact with the Earth by means of gravity, and with a knowledge of the force law I can compute the force I exort on the Earth and the force the Earth exerts on me given only a knowledge of our mutual relative coordinates in any coordinate system.

Later we will learn that the same is more or less true for the electromagnetic interaction – it is a lot more complicated, in the end, than gravity, but it is still true that a knowledge of the trajectories of two charged objects suffices to determine their electromagnetic interaction. and there is a famous paper by Wheeler and Feynman that suggests that even “radiation reaction” (something that locally appears as a one-body self-force) is really a consequence of interaction with a Universe of charge pairs.

This, then, allows us to cleanly differentiate real forces and pseudoforces. Real forces always involve two objects, where pseudolorces have no Newton’s Third Law partner! In the Elevator and Boxcar examples below, real gravity is identified by the fact that while the Earth pulls down on the mass in question, the mass pulls up on the Earth! Where the normal force acts on it, it pushes back on the object exerting the normal force. When tension in a string pulis it, it pulls back on the string.

There is no Newton’s third law partner to the F -ma pseudoforce arising from the acceleration of the frame! Furthermore, this “pseudogravity” behaves differently from actual gravity- it is (for example) perfectly uniform within the trame no matter how large the frame might be where actual gravity drops off (however slightly) as one moves verti cally, with the field lines slowly diverging (because the gravitational field diverges from its massive sources).

One has to work a bit harder to make this operational distinction clear when one builds a (relativistic, quantum) field theory, but throughout physics it remains the case that forces (or their quantum equivalents, “interactions”) never accur in isolation, while

pseudoforces “just happen” with nothing else making them happen.

The point is that in the end, the operational definition of an inertial frame is that it is a frame where Newton’s Laws are true for a closed, finite set of (force) Law of Nature that all involve well-defined interaction in the coordinates of the inertial frame. In that case we can add up all of the actual forces acting on any mass.

If the observed movement of that mass is in correspondence with Newton’s Laws given that total force, the frame must be inertial! Otherwise, there must be at least one “force” that we cannot identify in terms of any interaction pair, and examined closely, it will have a structure that suggests some sort of acceleration of the frame rather than interaction per se with a (perhaps still undiscovered) interaction law.

There is little more that we can do, and even this will not generally suffice to prove that any given frame is truly inertial. For example, consider the “rest frame” of the visible Universe, which can be thought of as a sphere some 13.7 billion Light Years⁶⁸ in radius surrounding the Earth.

To the best of our ability to tell, there is no compelling asymmetry of velocity or relative acceleration within that sphere-all motion is reasonably well accounted for by means of the known forces plus an as yet unknown force, the force associated with “dark matter” and “dark energy”, that still appears to be a local interaction but one we do not yet understand.

How could we tell if the entire sphere were uniformly accelerating in some direction, however? Note well that we can only observe near-Earth gravity by its opposition – in a freely falling box all motion in box coordinates is precisely what one would expect if the box were not falling! The pseudoforce associated with the motion only appears when relating the box coordinates back to the actual unknown inertial frame.

If all of this gives you a headache, well, it gives me a bit of a headache too. The point is once again that an inertial frame is practically speaking a frame where Newton’s Laws hold, and that while the coordinate frame of the visible Universe is probably the best that we can do for a Universal rest frame, we cannot be certain that it is truly inertial on a much larger length scale – we might be able to detect it if it wasn’t, but then, we might not.

Einstein extended these general meditations upon the invariance of frames to invent first special relativity (frame transformations that leave Maxwell’s Equations form invariant and hence preserve the speed of light in all inertial frames) and then general relativity, which is discussed a bit further below.

Let us consider a ray of light that shines through a window in an elevator at rest, as shown in figure. The ray of light follows a straight line path and hits the opposite wall of the elevator at the point P.

Let us now repeat the experiment, but let the elevator accelerate upward very rapidly, as shown in figure. The ray of light enters the window as before, but before it can cross the room to the opposite wall the elevator is displaced upward because of the acceleration. Instead of the ray of light hitting the wall at the point P, it hits at some lower point Q because of the upward acceleration of the elevator.

To an observer in the elevator, the ray of light follows the parabolic path, as shown in figure. Thus, in the accelerated coordinate system of the elevator, light does not travel in a straight line, but instead follows a curved path. But by the principle of equivalence the accelerated elevator can be replaced by a gravitational field. Therefore light should be bent from a straight line path in the presence of a gravitational field.

The gravitational field of the earth is relatively small and the bending cannot be measured on earth. However, the gravitational field of the sun is much larger and Einstein predicted in 1916 that rays of light that pass close to the sun should be bent by the gravitational field of the sun.

Another way of considering this bending of light is to say that light has energy and energy can be equated to mass, thus the light-mass should be attracted to the sun. Finally, we can think of this bending of light in terms of the curvature of spacetime caused by the mass of the sun. Light follows the shortest path, called a geodesic, and is thus bent by the curvature of spacetime.

Regardless of which conceptual picture we pick, Einstein predicted that a ray of light should be deflected by the sun by the angle of 1.75 seconds of arc. In order to observe this deflection it was necessary to measure the angular deviation between two stars when they are far removed from the sun, and then measure the deflection again when they are close to the sun. Of course when they are close to the sun, there is too much light from the sun to be able to see the stars.

Hence, to test out Einstein’s prediction it was necessary to measure the separation during a total eclipse of the sun. Sir Arthur Eddington led an expedition to the west coast of Africa for the solar eclipse of May 29, 1919, and measured the deflection. On November 6, 1919, the confirmation of Einstein’s prediction of the bending of light was announced to the world.

More modern techniques used today measure radio waves from the two quasars, 3c273 and 3c279 in the constellation of Virgo.

A quasar is a quasi-stellar object, a star that emits very large quantities of radio waves. Because the sun is very dim in the emission of radio waves, radio astronomers do not have to wait for an eclipse to measure the angular separation but can measure it at any time.

On October 8, 1972, when the quasars were close to the sun, radio astronomers measured the angular separation between 3c273 and 3c279 in radio waves and found that the change in the angular separation caused by the bending of the radio waves around the sun was 1.73 seconds of arc, in agreement with the general theory of relativity.