11.3: First Steps
- Page ID
- 57721
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)invariance off the interval gets us started
Recall that the coordinates \(y\) and \(z\) transverse to the direction of relative motion between rocket and laboratory have the same values in both frames (Section 3.6):
\[\] \[\begin{aligned} &y=y^{\prime} \\ &z=z^{\prime} \end{aligned}\]
where primes denote rocket coordinates. A second step makes use of the difference in observed clock rates when the clock is at rest or in motion (Section \(1.3\) and Box 3-3). Think of a sparkplug at rest at the origin of a rocket frame that moves with speed \(v_{\text {rel }}\) relative to the laboratory. The sparkplug emits a spark at time \(t^{\prime}\) as measured in the rocket frame. The sparkplug is at the rocket origin, so the spark occurs at \(x^{\prime}=0\).
Derive difference in clock rates
Where and when \((x\) and \(t)\) does this spark occur in the laboratory? That depends on how fast, \(v_{\text {rel }}\), the rocket moves with respect to the laboratory. The spark must occur at the location of the sparkplug, whose position in the laboratory frame is given by
\[x=v_{\mathrm{rel}} t \nonumber \]
Now the invariance of the interval gives us a relation between \(t\) and \(t^{\prime}\),
\[\left(t^{\prime}\right)^{2}-\left(x^{\prime}\right)^{2}=\left(t^{\prime}\right)^{2}-(0)^{2}=t^{2}-x^{2}=t^{2}-\left(v_{\mathrm{rel}} t\right)^{2}=t^{2}\left(1-v_{\mathrm{rel}}^{2}\right) \nonumber \]
from which
\[t^{\prime}=t\left(1-v_{\mathrm{re}}^{2}\right)^{1 / 2} \nonumber \]
or
\[t=\frac{t^{\prime}}{\left(1-v_{\mathrm{rel}}^{2}\right)^{1 / 2}} \quad \quad \quad \quad \text { [when } \mathrm{x}^{\prime}=0 \text { ] } \]
The awkward expression \(1 /\left(1-v_{\text {rel }}^{2}\right)^{1 / 2}\) occurs often in what follows. For simplicity, this expression is given the symbol Greek lower-case gamma: \(\gamma\).
\[\gamma \equiv \frac{1}{\left(1-v_{\mathrm{re}}^{2}\right)^{1 / 2}} \nonumber \]
Time stretch factor defined
Because it gives the ratio of observed clock rates, \(\gamma\) is sometimes called the time stretch factor (Section 5.8). Strictly speaking, we should use the symbol \(\gamma_{\text {rel }}\), since the value of \(\gamma\) is determined by \(v_{\text {rel }}\). For simplicity, however, we omit the subscript in the hope that this will cause no confusion. With this substitution, equation \(\PageIndex{2}\) becomes
\[t=\gamma t^{\prime} \quad \quad \quad \quad \text { [when } \mathrm{x}^{\prime}=0 \text { ] }\]
Substitute this into the equation \(x=v_{\text {rel }} t\) above to find laboratory position in terms of rocket measurements:
\[t=\gamma t^{\prime} \quad \quad \quad \quad \quad \text { [when } \mathrm{x}^{\prime}=0 \text { ] } \]
Equations \(\PageIndex{1}\), \(\PageIndex{3}\) and \(\PageIndex{5}\) give the first answer to the question, "If we know the space and time coordinates of an event in one free-float frame, what are its space and time coordinates in some other overlapping free-float frame?" These equations are limited, however, since they apply only to a particular situation: one in which both events occur at the same place \(\left(x^{\prime}=0\right)\) in the rocket