11.3: First Steps

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