What's the difference between Lorentz factor for frames and for particles?

[BomboshMan]

Lorentz factor for moving inertial reference frames is

λ = [itex]\frac{1}{\sqrt{1 - \frac{v²}{c²}}}[/itex],

where v is the relative velocity between the frames. But in my textbook (I'm only just learning relativity), it says the Lorentz factor for a particle is

λ_p = [itex]\frac{1}{\sqrt{1 - \frac{u²}{c²}}}[/itex],

where u is the velocity of the particle...but what's the difference between these two expressions?

The textbook introduces λ_p during the derivation of relativistic momentum, so I'm fine with where it came from, but then without explanation the book starts using it for time dilation Δt = λ_pΔ$\tau$. Little confused so would love it if someone could clear this up for me.

Thanks in advance!

 

[stevendaryl]

There are two different, but related, concepts:

1. Time dilation for a particle.
2. Time dilation for a coordinate transformation.

For particle time dilation: (For simplicity, let's just look at one spatial dimension)
If a particle travels along some trajectory described by an equation such as:

$x = f(t)$

where $(x,t)$ are coordinates in an inertial Cartesian coordinate system, then the proper time $\tau$ experienced by the particle as it travels along the trajectory is given by:

$\tau = \int \sqrt{1-\dfrac{v^2}{c^2}} dt$

where $v = \dfrac{df}{dt}$

For coordinate transformation time dilation: (Again, just using one spatial dimension): Suppose you have two inertial Cartesian coordinate systems $(x,t)$ and $(x',t')$, such that the spatial origin of the primed system moves at speed $v$ relative to the unprimed system. Let $e_1$ and $e_2$ be two events (points in spacetime). Let $(\delta(x),\delta(t))$ be their separation according to the unprimed coordinate system, and $(\delta(x'),\delta(t'))$ be their separation according to the primed coordinate system.

If

$\delta(x') = 0$

(so the two events take place at the same location, according to the primed coordinate system), then

$\delta(t') = \sqrt{1-\dfrac{v^2}{c^2}} \ \delta(t)$

Time dilation for particles is a physical effect, which is actually measurable. Time dilation for coordinate systems is just a property of a particular coordinate transformation (the Lorentz transformation). However, they are related, in the sense that the Lorentz transformation is what you would expect to apply if you create a coordinate system based on time-dilated clocks (so you can derive the LT from particle time dilation), and the other way around, if you assume that the laws of physics governing clocks is Lorentz-invariant and insensitive to past history of accelerations, then clock time dilation is derivable.