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'.

 

[sardar]

First of all, don't worry if you are feeling confused about these concepts. They can be quite complex and take some time to fully understand. Let me try to clarify the difference between the Lorentz transformation and time dilation equations.

The Lorentz transformation equation is used to calculate the time and space coordinates of an event in one reference frame, based on the coordinates of the same event in a different reference frame that is moving at a constant velocity relative to the first frame. This equation takes into account the effects of time and space dilation, as well as length contraction, which are all consequences of the special theory of relativity.

On the other hand, the time dilation equation is used to calculate the difference in time between two events that occur at the same location in one reference frame, but at different locations in another reference frame that is moving at a constant velocity relative to the first frame. This equation only takes into account the effect of time dilation, which means that time appears to pass slower for an observer in a moving reference frame compared to an observer in a stationary reference frame.

So, to answer your question, the t and t' in the Lorentz transformation equation represent the time coordinates of the same event in two different reference frames, while the t' in the time dilation equation represents the time coordinate of an event in a moving reference frame compared to the time coordinate of the same event in a stationary reference frame.

I hope this helps to clarify the difference between these two equations. Keep exploring and learning about these concepts, and don't hesitate to seek further clarification if needed. As a scientist, it's important to always strive for a deeper understanding of complex ideas.