What are 'inertial frames', or 'inertial frames of reference'? Indeed what are 'frames of reference' and 'inertia'? More importantly, are they in any way useful, and if so, how can we get hold of any?

Frames of Reference

Frames of reference are frameworks in which measurements are made. When you measure up for a new living room carpet, you are using your living room walls as part of your frame of reference, together with a tape measure. This is a good frame of reference because, should the carpet be fitted in six month's time, the measurements will still apply, despite your living room having moved 184,000,000 miles across the solar system; neither you, the carpet fitter, nor, least of all, the carpet itself cares where in the Universe your living room is, just so long as it can be found with the walls and floor intact.

Inertia

Inertia is that bizarre quality that all massive objects (read: 'objects that have mass' not just 'big objects') have, which is resistance to force. When the bus sets off suddenly and you find yourself unwittingly sitting on the back seat - that's your inertia. When astronauts are pinned to their seats during take-off - that's their inertia. The more mass an object has, the harder it is to shift. This is common sense¹.

Newton's Laws of Motion

It's also physics. The concepts of motion and inertia are embodied in a set of rules called Newton's Laws of Motion. These describe, in reasonably simple terms, how stuff moves, what it takes to get it to move, and what consequences can be expected as a result of moving it.

The first of Newton's laws of motion states that a body will remain at rest or in a state of uniform motion unless acted on by a force. Immediately, you should be asking yourself: 'What is meant by uniform motion?' This is a very complex and difficult question, and one which is pivotal to both explaining and defining the laws of motion. Let's assume you've asked the question anyway, so that this entry can proceed with its attempt to answer it.

Defining Uniform Motion

Imagine a scientist inside a mobile laboratory with no access to the outside world - OK, perhaps a mobile phone in case of emergencies; the point is the scientist never directly observes, listens to, or otherwise senses the outside world. Our scientist performs a simple experiment to predict the trajectory of a carefully controlled projectile. Now imagine the laboratory being moved along smooth rails. As long as the rails are straight and the speed is constant, the experiment will be identical to the first, but as soon as a corner is turned, or the speed changes, the results of the experiment change drastically: the projectile veers wildly off course, depending on the sharpness of the bend or the rate at which the speed changes. So, as long as the laboratory (our frame of reference) is either moving at the same speed in a straight line, or is stationary, the experiment has the same, predictable² result.

You can try this experiment yourself next time you go on a journey. Throw a ball straight up and catch it - as long as the vehicle is travelling constantly in a straight line, it'll be easy. Should the vehicle change direction, you'll start to have problems. Indeed, as soon as any acceleration³ at all, is introduced to our frame of reference, the laws of motion can no longer apply. Why? Because a projectile minding its own business suddenly appears to move of its own accord. What's really happening, of course, is the laboratory, or vehicle, which is being subjected to a force, is being accelerated away from it. We have now established what is meant by uniform motion: it is motion involving zero acceleration.

Defining Inertial Frames

Let us extend the capabilities of our mobile laboratory by building a window into it, and maybe some state-of-the-art GPS technology as well. The scientist now sees the rest of the world and can use that as a frame of reference. The random motions of the laboratory as well as the laboratory itself can be totally ignored, and the projectile's behaviour will be predicted by the laws of motion using the Earth as a frame of reference; we can use the laws of motion provided we are in a state of rest or uniform motion relative to the Earth.

This is what is meant by an 'inertial frame of reference' - any frame of reference in which an object behaves according to Newton's laws. These reference frames are crucial because it is only in these frames that Newton's laws of motion can be successfully applied. But isn't this a little arbitrary? What is so special about the Earth that, in our experiment above, it effectively defined these inertial frames and, now we understand their importance, how can we establish them precisely?

Locating Inertial Frames

This is a very necessary part of the laws of motion, for without an inertial frame we cannot predict the motion of anything very much. The search for inertial frames has had quite a history so far, and, in a delightfully basic, 18th Century physics mode, it starts with a bucket of water.

Absolute Space

An experiment was performed by Newton to observe the effects of rotating⁴ and also, therefore, non-inertial frames. He took a bucket full of water, attached it via the handle to a piece of rope and twisted it. The bucket was then spun and the untwisting of the rope kept it spinning. This is what he wrote about what he observed:

... the surface of the water will at first be level, just as it was before the vessel began to move; but subsequently the vessel, by gradually communicating its motion to the water, will make it begin sensibly to rotate, and the water will recede little by little from the middle and rise up at the sides of the vessel; its surface assuming a concave form.... At first when the relative motion of the water in the vessel was greatest, that motion produced no tendency whatever of recession from the axis, the water made no endeavour to move upwards towards the circumference, by rising at the sides of the vessel but remained level, and for that reason its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the rising of the water at the sides of the vessel indicated an endeavour to recede from the axis; and this endeavour reveals the real circular motion of the water, continually increasing till it had reached its greatest point, when relatively the water was at rest with the vessel....
- Isaac Newton, Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy)

This is describing, in a rather wordy way, a variation of what we have all observed on a day-to-day basis: when we stir our tea, we notice the liquid rise at the edges of the cup. What Newton proposed to account for non-inertial behaviour was a universal reference frame, a sort of grid that sat behind the whole of creation which we could measure things against. This was termed 'absolute space'. He also came up with the matching concept of 'absolute time' - a single clock which ticked away the seconds of the Universe. So, an inertial frame was simply a laboratory that was either stationary or moving uniformly relative to absolute space, not the Earth. Problem solved.

Mach's Principle

Well... no, not really, because all that did was to give the problem a name, rather than to provide any solution for it. We are left with no way of detecting absolute space at all. Even if we could test the laws of motion under every conceivable frame of reference - a formidable task for even the most dedicated and enthusiastic of physicists - all it would tell us is whether any particular frame was stationary or in a state of uniform motion relative to absolute space. Here's what a certain Austrian physicist by the name of Ernst Mach had to say about Newton's bucket experiment:

Newton's experiment with the rotating vessel of water simply informs us that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative motion with respect to the mass of the Earth and the other celestial bodies. No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass until they were several leagues thick.
- Ernst Mach, Die Mechanik in ihrer Entwicklung (The Science of Mechanics)

What Mach proposed was that a state of rest (or uniform motion) was defined by the overall mass distribution and motion of the 'Earth and the other celestial bodies'. The mass of the Universe interacted in some way with the mass of an item to give that item its inertia. This hypothesis has become known as 'Mach's Principle', and can be demonstrated by a simple experiment:

1. Stand outside on a starry night.
2. Notice your arms hang by your sides and the stars are stationary⁵.
3. Perform a pirouette.
4. Notice your arms are naturally drawn upwards, and the stars are spinning.

It would be something of a coincidence if, when all the stars appeared to be at rest, you were in an inertial frame, and when they moved you weren't. Surely it would be more natural to assume a state of rest or uniform motion was defined by how the Universe was behaving relative to you? After all, it's quite big, and it would be a little arrogant to assume it had no effect on you whatsoever. Going back to our moving laboratory, the effects of inertia on the projectile - in fact everything inside including the laboratory itself - can be attributed to the (relative) acceleration of the Universe at large⁶. Inertia manifests itself as the unwillingness of objects to detach themselves from what the rest of the Universe is doing, and an inertial frame is one which is at rest or in a state of uniform motion with the Universe as a whole. But, hang on a moment: Why does the Earth work as an inertial frame? It is, after all orbiting a sun orbiting the galaxy, which is, in turn..., in other words, it appears to be about as much of an inertial frame as our lurching laboratory.

A Little Bit of Relativity Theory

Einstein stated, in his postulates that led to the special theory of relativity, that the laws of physics are identical regardless of the motion of the inertial observer. In other words, the laws of physics in one inertial frame will have precisely the same outcome in any other inertial frame, as long as the observer, and the experiment, stay in a state of rest or uniform motion. Another way of looking at this is to say, given two arbitrary inertial frames, it is impossible to say which one is moving and which one is stationary - is the astronaut floating past the cosmonaut or vice versa, or are they both moving? - as there is no test that could be conducted to differentiate. Alternatively, we can turn this statement upside-down and say an inertial frame can be defined as a frame in which all the laws of physics work.

This should strike you as a profound, if not extraordinary, statement. Can the effect of acceleration really be as drastic as all that? Well, yes and no. In the above experiment the path of the projectile cannot be predicted effectively as the laboratory is being thrown about seemingly at random (before the budget was increased to allow for a window and GPS). Strictly speaking, the answer is 'Yes' because any experiment would be subjected to these random motions, rendering useless any attempts at prediction. The answer becomes 'No' should the scientist know about the path of the laboratory - the motions could be calculated in, as was the case when the Earth became the reference frame. The latter, however, should be regarded as cheating, as this requires knowledge of the outside world defeating the point of putting our scientist in a box in the first place. This, incidentally, is the reasoning behind introducing the arbitrary ('arbitrary' because they have no source) centrifugal and coriolis forces. The answer can also be 'No' for a particular law that is not altered in a non-inertial frame.

We now have a working model for determining an inertial frame. If all the laws of physics are obeyed, we have an inertial frame, otherwise we don't. The problem here is that we would like to know that our frame is inertial, without having to test every single law of physics, some of which haven't been discovered yet.

The Principle of Equivalence

There is, however, another way of tackling this problem of searching for inertial frames. It involves a major contributor to all the experiments outlined so far, and something we take so completely for granted it took the human race several hundred thousand years to realise it existed at all. Gravity. Can the laboratory ever be regarded as an inertial frame when we have gravity switched on all the time? This was the problem Einstein tackled when he first attempted to crowbar gravitational theory into his special theory of relativity.

Galileo showed, many years ago, that all objects fall at the same rate (ignoring air resistance). This odd, and somewhat counter-intuitive effect (only regarded as counter-intuitive because air resistance does so often play a part) can be rationalised as follows:

• Gravitational force felt by any object at any given distance from the Earth's surface is directly proportional to the mass of the object (the bigger it is, the stronger the force).

• The resistance to acceleration (the object's inertia) is also directly proportional to the mass of the object (the bigger it is, the harder it is to move).

• These two properties of a falling object cancel exactly, therefore an object always falls at the same rate, regardless of its mass.

A coincidence? Possibly. Einstein proposed that inertial mass and gravitational mass are, if not identical, at least equivalent - hence: the Principle of Equivalence⁷. This means a spaceship nowhere near any gravitation which is accelerating at (approximately) 9.8 metres per second every second would, if you were inside it, feel just like the surface of the Earth where objects also fall at a rate of (approximately) 9.8 metres per second every second. So, for our laboratory to be a truly inertial frame, it would need to be infinitely far away from any source of gravitation (tricky), or according to the Principle of Equivalence, freely falling towards a source of gravitation (expensive, but a lot easier). Does this explain how Mach's principle works? Or does it supplant, supersede or otherwise render totally irrelevant Mach's principle?

The Battle of the Principles

In order to determine our inertial frame, we need to know if we can use Mach's principle, the equivalence principle or whether they are, in fact, the same thing. In order to do this, we need some idea of how these principles differ (if at all) and how to test them.

Testing the Equivalence Principle

In determining an inertial frame using the equivalence principle, all we need to do is to make sure our laboratory is in free-fall. In other words, just set it drifting in space and let the sum of all the gravitational forces throughout the universe affect it freely. To test the equivalence principle, we need only to determine the difference between the gravitational and inertial masses of a body. Newton himself tested for these differences by using a pendulum with weights of different constitution. In fact, all experiments to date have shown no differences at all to an accuracy of 1 part in 10¹¹⁸. This represents a difference of less than one gramme in 100,000 metric tonnes and, although this does not explicitly prove the equivalence principle, it does at least go a long way to backing it up.

Testing Mach's Principle

Mach's principle is rather more difficult to prove or disprove, as it does not specify to what degree any particular celestial body has an influence on another's inertia. What we can say, however, is that over mediocre cosmic distances (say, a few hundred-million light-years), the Universe looks pretty much the same; there are an inconceivably ludicrous number of galaxies in every direction. Given every celestial body is in some way responsible for the effect of Mach's principle, all we need to do is find a suitably large body that's not so far it gets swamped by the vast cosmic quagmire, yet far enough that it can be sufficiently large and - we hope - cosmically significant.

There is one body that satisfies these criteria quite nicely: our own Milky Way. As we are on the outskirts of this enormous swirl of matter, we can consider ourselves (mostly) outside of it and this local asymmetry ought to be detectable if Mach's principle is correct. The only problem is how accurate we need to be before we can detect it. The experiments that have been performed to detect this difference found none to an accuracy of 1 part in one hundred million trillion (1:10²⁰).

Yes, But What Does It All Mean?

Inertial frames are fundamental to many of the laws of physics. Yet, although the experimentation to date points to Einstein's principle of equivalence as being the main contender, the laws of physics are sufficiently vague to prevent us from knowing for certain whether or not we have a valid frame of reference in which to test them. This is quite a tricky situation, and it will take a huge leap of knowledge before we can, with any degree of certainty, measure up for our living room carpets.

Bibliography

• Concepts of Modern Physics 1994, Arthur Beiser (McGraw-Hill)
• Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity 1972, Steven Weinberg (John Wiley & Sons)
• Principles of Cosmology and Gravitation 1989, M. V. Berry (Adam Hilger)