also as before. Now we get T directly from the one dimensional statics problem along the yˆ′ direction:

as – naturally – before. We get the same answer either way, and there isn’t much di erence in the work required. I personally prefer to think of the problem, and solve it, in the inertial ground frame, but what you experience riding along in the boxcar is much closer to what the second approach yields – gravity appears to have gotten stronger and to be pointing back at an angle as the boxcar accelerates, which is exactly what one feels standing up in a bus or train as it starts to move, in a car as it rounds a curve, in a jet as it accelerates down the runway during takeo .

Sometimes (rarely, in my opinion) it is convenient to solve problems (or gain a bit of insight into behavior) using pseudoforces in an accelerating frame (and the latter is certainly in better agreement with our experience in those frames) but it will lead us to make silly and incorrect statements and get problems wrong if we do things carelessly, such as call mv2/r a force where it is really just mac, the right and side of Newton’s Second Law where the left hand side is made up of actual force rules. In this kind of problem and many others it is better to just use the real forces in an inertial reference frame, and we will fairly religiously stick to this in this textbook. As the next discussion (intended only for more advanced or intellectually curious students who want to be guided on a nifty wikiromp of sorts) suggests, however, there is some advantage to thinking more globally about the apparent equivalence between gravity in particular and pseudoforces in accelerating frames.

2.4.2: Advanced: General Relativity and Accelerating Frames

As serious students of physics and mathematics will one day learn, Einstein’s Theory of Special Relativity70 and the associated Lorentz Transformation71 will one day replace the theory of inertial “relativity” and the Galilean transformation between inertial reference frames we deduced in week 1. Einstein’s result is based on more or less the same general idea – the laws of physics need to be invariant under inertial frame transformation. The problem is that Maxwell’s Equations (as you will learn in detail in part 2 of this course, if you continue) are the actual laws of nature that describe electromagnetism and hence need to be so invariant. Since Maxwell’s equations predict the speed of light, the speed of light has to be the same in all reference frames!

This has the consequence – which we will not cover in any sort of detail at this time – of causing space and time to become a system of four dimensional spacetime, not three space dimension plus time as an independent variable. Frame transformations nonlinearly mix space coordinates and time as a coordinate instead of just making simple linear tranformations of space coordinates according to “Galilean relativity”.

Spurred by his success, Einstein attempted to describe force itself in terms of curvature of spacetime, working especially on the ubiquitous force of gravity. The idea there is that the pseudoforce produced by the acceleration of a frame is indistinguishible from a gravitational force, and that a generalized frame transformation (describing acceleration in terms of curvature of spacetime) should be able to explain both.

This isn’t quite true, however. A uniformly accelerating frame can match the local magnitude of a gravitational force, but gravitational fields have (as we will learn) a global geometry that cannot be matched by a uniform acceleration – this hypothesis “works” only in small volumes of space where gravity is approximately uniform, for example in the elevator or train above. Nor can one match it

70Wikipedia: http://www.wikipedia.org/wiki/Special Relativity.

71Wikipedia: http://www.wikipedia.org/wiki/Lorentz Transformation.

with a rotating frame as the geometric form of the coriolis force that arises in a rotating frame does not match the 1/r2rˆ gravitational force law.

The consequence of this “problem” is that it is considerably more di cult to derive the theory of general relativity than it is the theory of special relativity – one has to work with manifolds72 . In a su ciently small volume Einstein’s hypothesis is valid and gives excellent results that predict sometimes startling but experimentally verified deviations from classical expectations (such as the precession of the perihelion of Mercury)73

The one remaining problem with general relativity – also beyond the scope of this textbook – is its fundamental, deep incompatibility with quantum theory. Einstein wanted to view all forces of nature as being connected to spacetime curvature, but quantum mechanics provides a spectacularly di erent picture of the cause of interaction forces – the exchange of quantized particles that mediate the field and force, e.g. photons, gluons, heavy vector bosons, and by extension – gravitons74 . So far, nobody has found an entirely successful way of unifying these two rather distinct viewpoints, although there are a number of candidates75 .

72Wikipedia: http://www.wikipedia.org/wiki/Manifold. A manifold is a topological curved space that is locally “flat” in a su ciently small volume. For example, using a simple cartesian map to navigate on the surface of the “flat” Earth is quite accurate up to distances of order 10 kilometers, but increasingly inaccurate for distances of order 100 kilometers, where the fact that the Earth’s surface is really a curved spherical surface and not a flat plane begins to matter. Calculus on curved spaces is typically defined in terms of a manifold that covers the space with locally Euclidean patches. Suddenly the mathematics has departed from the relatively simple calculus and geometry we use in this book to something rather di cult...

73Wikipedia: http://www.wikipedia.org/wiki/Tests of general relativity. This is one of several “famous” tests of

the theory of general relativity, which is generally accepted as being almost correct, or rather, correct in context.

74Wikipedia: http://www.wikipedia.org/wiki/gravitons. The quantum particle associated with the gravitational field.

75Wikipedia: http://www.wikipedia.org/wiki/quantum gravity. Perhaps the best known of these is “string theory”,

but as this article indicates, there are a number of others, and until dark matter and dark energy are better understood as apparent modifiers of gravitational force we may not be able to experimentally choose between them.