Time dilation and Minkowski diagram

[Fredrik]

If 1' is supposed to be the event on the time axis of S' that has the same time coordinate in S' as 1 has in S, then the events 1 and 1' will be on the same hyperbola $-(ct^2)+x^2=-(c\Delta t)^2$. I don't know what formula you used to draw the curve in your diagram, but it clearly isn't that hyperbola (since it intersects the x=ct line).

Note that for any point (ct,x) on that hyperbola with x,t>0, we have $ct=\sqrt{x^2+(c\Delta t)^2}>x$, so the hyperbola must be drawn above the 45° line. Also note that $$\frac{ct}{x}=\sqrt{1+\frac{(c\Delta t)^2}{x^2}}\to 1$$ as $x\to\infty$. So the hyperbola will get closer and closer to the 45° line as x grows, but never intersect it.

 

[pervect]

Why are these lines and points so important for me to include in the picture?

The lines are the graphical illustration of "Ziga thinks Rinja's clock is running slow" and "Rinja thinks Ziga's clock is running slow". I.e .the diagram represents the concept of time dilation.

Your diagram as is doesn't contain those concepts, alas. I'm not quite sure what you think your diagram IS proving.

I originally wrote a lot more, but I think it was too wordy, so I deleted most of it.

 

[Jeronimus]

If i add passive transformation along the active one i get the distance which i wrote as Δt′′. What is this? It is not the same value as my Δt′ which i got using the active transformation.

Here is the picture. I only added a line parallel to x′ axis and marked a distance Δt′′.

Alright, after getting confused myself because i have not done such minkowski diagrams for a long time, here are my 2 cents.

I call the diagram crossing E0/E0' which is the synced reference point.

Accordingly, the time-interval measured by Ranja between 0' and 1' on your diagram, would be a clock Ranja carries and which measures 1 second between those two event point, E0' and let's say E1'.

I would label the time interval between E0'/E1' as Δt ( you did not label this one at all).

So you can directly apply the formula as wikipedia has it.

Δt' = Δt * γ (according to my labels)

As you can see in the diagram, the time interval between E0'/E1' is measured to be longer than 1 second seen from Ziga's point of view.

Now let's check the interval between 0/1 in your diagram. Let's call those events E0/E1. A clock Ziga carries, would measure 1 second between E0/E1.

I would label this time-interval as Δt2 (you labeled it Δt). The time interval you labeled as Δt'' i would label as Δt2'.
Again, as you can see, while Ziga measures this time interval between E0/E1 to be 1 second, Ranja measures this as Δt2' > 1 second. Again, the same formula applies

Δt2' = Δt2 * γ (according to MY labels)

The parallel line to x' you drew, which goes through 1 and crosses Ranja's time axis, is equivalent to the parallel line to the x-axis which goes through 1' and crosses Ziga's time axis.
The cross-points allow to directly see the time-intervals as measured by the observer at rest. (i am too rusty on such diagrams to understand how minkowski did this magic)

This is another great achievement of genius Minkowki, allowing us to draw two diagrams within one space overlaping in such a way that we can extract information about time dilation and length contraction without actually having to do the maths.

 

[phyti]

Here is a variation of the spacetime drawing that eliminates the hyperbola and provides an observer perspective resulting from time dilation.
On the left, an event e occurs at U(t,x)=(1,1).
Observer A, moving at .6c relative to U, intercepts the reflected light from e at event 6, at U(1.25,.750).
An arc with r=1.25 intersects a vertical line from 6.
That point is projected to the ct axis, giving A(t,x)=(1.00,.600). A thinks event e occurred at e6.
The right figure uses the same method for .3c and .9c.
If you have any CAD experience, the software can do the calculations.

https://www.physicsforums.com/attachments/52325

 

[71GA]

Alright, after getting confused myself because i have not done such minkowski diagrams for a long time, here are my 2 cents.

I call the diagram crossing E0/E0' which is the synced reference point.

Accordingly, the time-interval measured by Ranja between 0' and 1' on your diagram, would be a clock Ranja carries and which measures 1 second between those two event point, E0' and let's say E1'.

I would label the time interval between E0'/E1' as Δt ( you did not label this one at all).

So you can directly apply the formula as wikipedia has it.

Δt' = Δt * γ (according to my labels)

As you can see in the diagram, the time interval between E0'/E1' is measured to be longer than 1 second seen from Ziga's point of view.

Now let's check the interval between 0/1 in your diagram. Let's call those events E0/E1. A clock Ziga carries, would measure 1 second between E0/E1.

I would label this time-interval as Δt2 (you labeled it Δt). The time interval you labeled as Δt'' i would label as Δt2'.
Again, as you can see, while Ziga measures this time interval between E0/E1 to be 1 second, Ranja measures this as Δt2' > 1 second. Again, the same formula applies

Δt2' = Δt2 * γ (according to MY labels)

The parallel line to x' you drew, which goes through 1 and crosses Ranja's time axis, is equivalent to the parallel line to the x-axis which goes through 1' and crosses Ziga's time axis.
The cross-points allow to directly see the time-intervals as measured by the observer at rest. (i am too rusty on such diagrams to understand how minkowski did this magic)

This is another great achievement of genius Minkowki, allowing us to draw two diagrams within one space overlaping in such a way that we can extract information about time dilation and length contraction without actually having to do the maths.

I understand this now.

Here is a variation of the spacetime drawing that eliminates the hyperbola and provides an observer perspective resulting from time dilation.
On the left, an event e occurs at U(t,x)=(1,1).
Observer A, moving at .6c relative to U, intercepts the reflected light from e at event 6, at U(1.25,.750).
An arc with r=1.25 intersects a vertical line from 6.
That point is projected to the ct axis, giving A(t,x)=(1.00,.600). A thinks event e occurred at e6.
The right figure uses the same method for .3c and .9c.
If you have any CAD experience, the software can do the calculations.

https://www.physicsforums.com/attachments/52325

Could you please mark axis in your picture? Picture would make more sense to me then. :)