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

• BomboshMan
In summary, the Lorentz factor for particle time dilation is given by λp = \frac{1}{\sqrt{1 - \frac{u2}{c2}}}, where u is the velocity of the particle. This is used in the derivation of relativistic momentum and also in the time dilation equation Δt = λpΔ\tau. The Lorentz factor for coordinate transformation time dilation is given by λ = \frac{1}{\sqrt{1 - \frac{v2}{c2}}}, where v is the relative velocity between the frames. These two concepts are related but are different in their applications and origins.
BomboshMan
Lorentz factor for moving inertial reference frames is

λ = $\frac{1}{\sqrt{1 - \frac{v2}{c2}}}$,

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

λp = $\frac{1}{\sqrt{1 - \frac{u2}{c2}}}$,

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

The textbook introduces λp 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 = λpΔ$\tau$. Little confused so would love it if someone could clear this up for me.

Last edited:
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.

## 1. What is the Lorentz factor for frames?

The Lorentz factor for frames is a mathematical factor used in the special theory of relativity to describe the relationship between time and space in different reference frames. It is denoted by the symbol γ (gamma) and is equal to 1/√(1-v²/c²), where v is the velocity of the frame and c is the speed of light.

## 2. What is the Lorentz factor for particles?

The Lorentz factor for particles is a similar concept, but it describes how the properties of a particle, such as mass and energy, change in different reference frames. It is also denoted by the symbol γ and is equal to 1/√(1-v²/c²), where v is the velocity of the particle and c is the speed of light.

## 3. What is the difference between the Lorentz factor for frames and for particles?

The main difference is that the Lorentz factor for frames describes the relationship between time and space in different reference frames, while the Lorentz factor for particles describes the change in properties of a particle in these frames. Additionally, the Lorentz factor for frames is associated with the observer's perspective, while the Lorentz factor for particles is associated with the particle's perspective.

## 4. How is the Lorentz factor related to the theory of relativity?

The Lorentz factor is a key concept in the special theory of relativity, which states that the laws of physics are the same for all inertial reference frames. It is used to calculate how time, space, and other physical quantities are perceived differently in different frames of reference, and is crucial in understanding the effects of time dilation and length contraction.

## 5. How does the Lorentz factor affect our understanding of the universe?

The Lorentz factor plays a crucial role in our understanding of the universe and has led to groundbreaking discoveries, such as the concept of time dilation and the famous equation E=mc². It also helps explain phenomena such as the constancy of the speed of light and the concept of spacetime. Without the Lorentz factor, our understanding of the universe would be limited and many important scientific advancements would not have been possible.

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