rgtr said:
Summary: Why do I divide T_moving by T_stationary for light clocks and not minus T_moving - T_stationary?
In special relativity I can get ## \gamma ## , ## \frac {T_B}{T_A}=\gamma ## Why do I not go ##{T_B} - {T_A} = \gamma## ?
##T_B = \frac {2H} {c^2 - v^2}## . ## T_B ## is the moving light clock.
## T_A = \frac {2H} {c^2} ## . ##T_A ## is the stationary light clock
[snip]
I am using this book
https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf page 18 EQ (1.11)
Thanks
rgtr said:
Why does it need to be a function of v? And what do you mean by a function of v?
One can try to define whatever one wants along as it's consistent.
One then hopes the definition is useful.\begin{align*}
\frac{T_B}{T_A} \stackrel{(1.11)}{=}\frac{c}{\sqrt{c^2-v^2}}=\frac{1}{\sqrt{1-v^2/c^2}}\stackrel{(1.12)}{\equiv}\gamma
\end{align*}
morin-p19 said:
The ##\gamma## factor here is ubiquitous in special relativity
So, ##\gamma## is already defined and is (according to Morin) ubiquitous [and presumably, useful]
... and is called the ##\gamma##-
factor (
not ##\gamma##-
difference).
It is dimensionless since it is the ratio of two times.
##\gamma\equiv\frac{1}{\sqrt{1-v^2/c^2}}## is a "function of ##v## alone" (since ##c## is a constant) and "not a function of ##H##".
That is, the ##\gamma##-factor depends on the relative-velocity ##v## and not on the size of transverse light-clock. Doubling ##H## will double ##T_B## and double ##T_A##, but not their ratio ##\gamma##.
You can try to define the difference of times, but you can't use ##\gamma## since it's already taken.
Let's call it ##\Delta T \equiv T_B-T_A##, the time-
difference
\begin{align*}
{T_B}-{T_A} = \frac{2H}{\sqrt{c^2-v^2}} - \frac{2H}{c} =\frac{2H}{c}\left( \frac{1}{\sqrt{1-v^2/c^2}}-1\right) \equiv \Delta T
\end{align*}
##\Delta T## is a function of ##v##
and ##H##.
It has dimensions of a time since it is the difference of two times.
Doubling ##H## will double ##T_B## and double ##T_A##, and thus double the difference ##\Delta T##.
It is consistent and has its uses in various situations, but (it turns out) is not as useful as ##\gamma## in studying various aspects of relativity. (Later, you see that it's needed in studying the Michelson-Morley experiment. It could be helpful in the study of the Doppler factor. The energy analogue of the difference is helpful for relativistic kinetic energy.)
It turns out a lot special relativity reduces to studying
triangles on a position-vs-time graph (called a spacetime diagram)
and one often deals with the analogue of a right-triangle in the Euclidean plane.
In Euclidean circular trigonometry, the ratio of sides is more useful that the difference of sides.
The ratio of sides tells us (via various definitions and formulae) something about the relative-slope
and the angle between the sides. Doubling the size of the triangle doesn't change the slope or the angle.
In relativity (using hyperbolic trigonometry), the relative-velocity is the analogue of the slope.