Lorentz Transformation and Time Dilation

[SteveDC]

I've managed to confuse myself and don't understand the difference between the formula for Lorentz time transformation (t'=γ(t-vx/c^2) and the time dilation equation t'=γ(t_proper)

As I understand, proper time is difference between two events that happen in same place in a given reference frame.

So what do the t's relate to in the Lorentz transformation equation as they must be something different?

 

[jtbell]

In the Lorentz transformation equations, t is the time coordinate of a single event in one inertial reference frame (IRF). t' is the time coordinate of the same event in a different IRF, that is moving with respect to the first one with velocity v.

In the time dilation formula, t is the time interval between two events that occur at the same location in one IRF, and t' is the time interval between the same two events in a different IRF, that is moving with respect to the first one with velocity v.

 

[SteveDC]

Makes sense - Thanks

 

[yuiop]

We can write the transformation for the interval between two events like this:

Δt'=γ(Δt-vΔx/c^2)

if Δt is the interval between two events at the same location in S, then xΔ=0 and the time interval measured in S' is then:

Δt'=γ(Δt)

Δt is the proper time measured in S and Δt' is the coordinate time measured in S'. However, if we calculate the transformation to S' of the spatial interval between the same two events we get:

Δx'=γ(Δx-vΔt)

Since we defined Δx as zero in S, and if Δt is non zero, then we get:

Δx'=γ(vΔt) ≠ 0

which means the two events do not occur in the same place in S' and that is why it is called a coordinate time interval and not a proper time interval in S'.