A Space-time Coordinate System: The Lorentz Transformations (Time)
The Lorentz Transformation (Part 2)
In the previous post, we derived the equation telling us that if we know when and where an event happens (x,t); we can work out where it happens for an observer moving at speed v (x').
So what about when an event happens?
Firstly we need to consider simultaneity. We know clocks in the front of motion lag behind clocks towards the back (think back to the moving train where light hit the back first). This means according to team platform; all of team train's clocks are out of sync and vice versa.We can see this clearly if we think about how the teams synchronise their clocks.
When the origins of both frames cross; both members at the origin send out a flash of light. This simply means at x=0, t=0 and at x'=0, t'=0. This means that the origins of both frames are synchronised. Members along the lines know how far they are from the origin. This means when they see the light, they can work out how long ago the flash was emitted by doing time=distance/speed. If they know how long ago the clock at the origin was set to 0; all they need to do is add the time it took for the light to reach them to their clock and then start their clock. This way all members are synchronised.
But from team platform's POV team train is moving to the right. This means people towards the right are always moving away from where the light was emitted and people to the left are always moving towards where the light was emitted.
Team platform says team train didn't consider movement and so are out of sync.
Along with this, team platform see the distance from the origin to the members as length contracted and see the clocks of all the members ticking slowly.
This assumes all the motion is along the x plane and so the full transformations would also include
y = y'
z = z'
No motion occurs in the x or y planes and so nothing changes.
There are also Lorentz transformations for 3D spaces. These are more complicated and involve splitting the velocity of the moving observer into x, y and z components.
You should also remember that it is valid to take any inertial frame of reference as stationary. If we took team train as stationary the equations would be the same expect we swap normal coordinates for primed coordinates and we change the sign of the velocity.
There is also a more symmetric form of the equation to make them look more like each other.
Multiplying the time transformation by c and putting c/c into the space transformation
ct' = γ(ct - x(v/c)
x' = γ(x - ct(v/c))
y' = y
z' = z
These are sometimes used as they look nicer. Note that ct is a speed x time and so is a distance.
In the derivation we used x, x', t and t' to show the difference between the those points and the origin (since this was where the clocks were synchronised).
In situations where we are not calculating from the origin, we use the change in (Δ/delta) x, x', t and t'.
For example if we wanted to know the difference between x = 10 and x = 5. You work out Δx
= 10 - 5
= 5
We may therefore also write the equations as the following for situations where we are not looking at the distance from the origin.
Δt' = γ (Δ t- vΔx/c2)
Δx' = γ (Δx - vΔt)
Now if we have any event in space-time. We can say where it occurs and when it occurs according to all observers using the Lorentz transformations.
Thanks for reading. If you enjoyed this post or any of my others, follow and subscribe to my blog. Feel free to discuss anything related to this post or ask questions in the comments below.
Check out my other posts on special relativity! (Link to all the posts)
How To Travel To The Future
How Galilean Relativity And The Lorenz Maxwell Equations (The 2 Postulates) Contradict
The Postulates Of Special Relativity
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