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2. Galilean Relativity

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  • [ "article:topic", "Chapters", "Galilean relativity", "authorname:knoxl" ]

    We now extend our discussion of spatial geometry to spacetime geometry. We begin with Galilean relativity, which we will then generalize in the next section to Einstein (or Lorentz) relativity.

    The notion of relativity of motion is not something new with Einstein. It's built into Newtonian mechanics as well. The idea is that the results of experiments done in any inertial frame should be the same; i.e., one should not be able to determine by some experimental means what frame is at "absolute rest" and which frames are moving -- all motion is relative. An inertial frame is one in which Newton's laws of motion are satisfied. These laws prescribe accelerations, not velocities, so a frame that is moving at constant relative velocity with respect to an inertial frame must itself be an inertial frame. The principle of Galilean relativity can be stated as follows: 

    If the unprimed coordinates are Euclidean coordinates specifying an inertial frame then the primed ones given by

            t' & = t, \\ x' &= x - vt, \\ y' &= y,\ {\rm and}\\ z' & = z

    are as well. Because the relationship is time dependent (and because we are anticipating the generalization to Lorentz transformations), we have explicitly included time in the transformation, even though time transforms trivially. We will call such a transformation a "Galilean Boost." In this case the above equations describe the relationship between a primed reference frame and an unprimed reference frame that is moving relative to it with speed \(v\) in the \(+x\) direction.

    One can quickly verify that, if evaluated at the same time \(t\), the distance between two infinitesimally separated points is unchanged by this transformation. Further, the transformation is symmetric. That is, there is nothing special about the primed frame relative to the unprimed frame. The reverse transformation, found by solving the above for \(t\), \(x\), \(y\), and \(z\) is 

            t & = t', \\ x &= x' + vt', \\ y &= y',\ {\rm and}\\ z & = z'

    which is the same equation as above except with the replacement of \(v\) with \(-v\). That is, it is exactly the same rule, with \(v \rightarrow -v\) because while the primed frame is moving relative to the unprimed frame toward higher \(x\), the unprimed frame is moving relative to the primed frame toward lower \(x'\).

    Box \(\PageIndex{1}\)

    Exercise 2.1.1: With the Galilean boost transformation, velocities add in a simple manner.  If \(u' = dx'/dt\) where \(x'(t)\) is the \(x'\) location of some particle, find \(u = dx/dt\) as a function of \(u'\) and \(v\).

    Box \(\PageIndex{2}\)

    Exercise 2.2.1: Show that Newton's Law for a spring with an equilibrium length of zero, spring constant \(k\), centered at \(x=x_c\), \( -k (x-x_c)   = m {\ddot x}\), is invariant under the Galilean transformation.

    Box \(\PageIndex{3}\)

    Exercise 2.3.1: Discuss what this invariance means for applicability of Newton's laws in both the primed and unprimed frames, and the question: "is such a law consistent with the principal of Galilean relativity?"