Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

11.3: First Steps

( \newcommand{\kernel}{\mathrm{null}\,}\)

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):

y=yz=z

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 vrel  relative to the laboratory. The sparkplug emits a spark at time t as measured in the rocket frame. The sparkplug is at the rocket origin, so the spark occurs at x=0.

Derive difference in clock rates

Where and when (x and t) does this spark occur in the laboratory? That depends on how fast, vrel , 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=vrelt

Now the invariance of the interval gives us a relation between t and t,

(t)2(x)2=(t)2(0)2=t2x2=t2(vrelt)2=t2(1v2rel)

from which

t=t(1v2re)1/2

or

t=t(1v2rel)1/2 [when x=0 ] 

The awkward expression 1/(1v2rel )1/2 occurs often in what follows. For simplicity, this expression is given the symbol Greek lower-case gamma: γ.

γ1(1v2re)1/2

Time stretch factor defined

Because it gives the ratio of observed clock rates, γ is sometimes called the time stretch factor (Section 5.8). Strictly speaking, we should use the symbol γrel , since the value of γ is determined by vrel . For simplicity, however, we omit the subscript in the hope that this will cause no confusion. With this substitution, equation 11.3.2 becomes

t=γt [when x=0 ] 

Substitute this into the equation x=vrel t above to find laboratory position in terms of rocket measurements:

t=γt [when x=0 ] 

Equations 11.3.1, 11.3.3 and 11.3.5 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 (x=0) in the rocket


This page titled 11.3: First Steps is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Edwin F. Taylor & John Archibald Wheeler (Self-Published (via W. H. Freeman and Co.)) via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?