Online Astronomy eText: Background Physics: Motion and Forces
Inertial and Non-Inertial Reference Frames
(considerable overlap with Fictitious Forces)

A Brief Review of Newton's Laws of Motion
      Let's review certain basic concepts of motion, namely Newton's first two Laws of Motion, which are presumably as basic and fundamental as any natural law can be:
      (1) The Law of Inertia: A body which has no force acting on it will move with uniform motion (that is, with constant speed and direction).
      (2) The Force Law: If a force acts on a body, it will not move uniformly, but will be accelerated in the direction of the force at a rate proportional to the force, and inversely proportional to its inertia, or mass.
      Now, these two laws seem very simple and obvious, and perfectly reasonable and correct. So much so, that if we see an object which is moving uniformly, we presume that it must not have any force (or at least, any net force) acting on it; whereas if we see an object which is accelerating, we presume it must have some force acting on it, in the direction of its acceleration. The strange thing is, that it is not only very easy, but actually more normal than not, for Newton's Laws of Motion to be wrong. For we often find ourselves in a situation in which bodies appear to be accelerating under the influence of some force, even though no such force is actually acting on them.

Inertial Frames of Reference
      To understand how such a statement could possibly be true, we need to discuss frames of reference. A frame of reference is simply that portion of the world around us, which we use to measure the motion of moving bodies. For all practical purposes, the world around us appears to be at rest, and insofar as that statement is true, then any motion we measure relative to our surroundings is correctly observed, and if a motion appears uniform, it must truly be uniform, and if the motion appears nonuniform, then it must truly be nonuniform.
      But suppose that instead of using the world around us, we use some particular portion of the world, such as a railway car, which is moving relative to the rest of the world. As the car moves along its tracks, the motion of whatever object we are observing will not be measured correctly, but will have an error equal to the motion of the railway car. So wouldn't that affect our observations of the moving object? Of course it would; but as long as the motion of the railway car is absolutely uniform and unchanging, whatever error we make in observing the moving object will be absolutely constant and unchanging, as well; so if the object has uniform motion in the everyday world that is our normal frame of reference, then it will appear to have a different but still absolutely uniform motion in the frame of reference represented by the railway car. And if the moving object is moving nonuniformly, because there is a force acting on it, in the everyday world, then it will also be moving nonuniformly, by exactly the same amount, in the frame of reference of the moving railway car.
     This concept, that a uniformly moving frame of reference, such as the railway car, should not change the laws of motion, was actually first proposed, albeit in a slightly different form, decades prior to Newton's stating his Laws, by Galileo Galilei. Galileo proposed that in all frames of reference which are moving uniformly relative to each other, the laws of nature must be the same. This statement encompasses not only Newton's Laws, but all the laws of nature, and is the basis of what we call Galilean relativity.
      Now, let's suppose that in the everyday world, Newton's Laws of Motion are correct, and most particularly, that the Law of Inertia is correct. If so, then in any frame of reference that is moving absolutely uniformly relative to the everyday world, the Law of Inertia will still be correct. All frames of reference, in which the Law of Inertia is correct, are called inertial frames. Frames of reference in which the Law of Inertia is not correct, are called non-inertial frames.

Non-Inertial Reference Frames
     But how in the world could you possibly find yourself in such a situation that the Law of Inertia would appear to be wrong? Very easily; for remember, in discussing the uniformly moving railway car, it was stated to be moving uniformly, so that any observations of a moving object would have an error, relative to the everyday world, but an error that was as constant as the motion of the railway car, so that uniform motion still appeared uniform, and nonuniform motion still appeared nonuniform.
      But what if the motion of the railway car is not constant? Then the error introduced into measurements of the moving object would be changing with time, which would make a constant motion look nonuniform, and therefore accelerated, which would suggest that a force is acting on the moving object, which is obviously incorrect, if it is really moving with constant motion. In other words, if your frame of reference has a non-uniform, or accelerated motion, then the Law of Inertia will appear to be wrong, and you must be in a non-inertial frame of reference. So, although all frames of reference which are moving uniformly relative to an inertial reference frame are also inertial reference frames, all frames of reference which are moving non-uniformly (are accelerated) relative to an inertial reference frame are non-inertial reference frames.

D'Alembert, or "Fictitious" Forces
     So, what are some examples of non-inertial reference frames? And what are the consequences of being in a non-inertial reference frame, as opposed to being, as we usually presume we are, in an inertial reference frame?