# Time dilation and Minkowski diagram

pervect
Staff Emeritus
After i figured out how to show length contraction in this topic. I tried to use a similar way to show time dilation in Minkowski diagram. Time dilation means that time interval between two events is the shortest in the frame in which those two events happen in same place. We call this frame its rest frame.
Hmm - there's a problem here already, as the inertial observer experiences the longest proper time between events,

You are missing at least two lines on your diagram, and I don't see them in in any subsequent posts, so I'm guessing that you're still missing something. I'd suggest adding three more lines, and two more points:

One important pre-requisite, which I'll call PRE Make sure that the Lorentz interval, also called the proper time, between the origin and point 1 is the same as the Lorentz interval between the origin and point 1'. I think you addressed this by saying that points 1 and 1' should both be on the same hyperbola of (ct)^2 - x^2 = constant.

The first line that you are missing is the set of points that are simultaneous with event 1 in the (t,x) frame. Hint: this set of points is parallel to the x axis.

The second line is the set of points that are simultaneous with event 1 in the (t', x') frame. Hint: this set of points is parallel to the x' axis. The important thing to note is that this set of points is DIFFERENT that the set of points of our first line. Otherwise, we won't be using this line.

The third line that you are missing is the set of points that are simultaneous with event 1' in the (t', x') frame. Hint: this is parallel to the x' axis, similar to line 2, but not similar to line 1.

So those are the three lines I suggest you add - now come the points, and the meaning of time dilation on your graph.

In the (t,x) frame, line 1 represents the set of points simultaneous with the event you labelled as "1". Find point 2, where this line intersects Ranja's worldline. Time dilation says that point if point 1' occurs after point 2, because using the simultaneity conventions of the (t,x) frame, the clock in the (t', x') frame is ticking slowly. (Be sure that the precondition PRE is met!)

In the (t', x') frame, line 3 represents the set of points simultaneous with the event you labelled as 1'. Find point 3, where this line intersects Ziga's worldline. Time dilation says that point 1 comes after point 3. Again, be sure the precondition PRE is met.

"Numerical value of ##\Delta t''##" as measured in frame ##x'ct'## or as measured in ##x,ct##?
##\Delta t''## is the time coordinate of the point ##1## with respect to the ##S'## frame.

It is not a real hyperbola, so maybe this is the case. Could anyone provide a picture which proves this?
I personally cannot, and while the diagrams are handy for seeing things, at some point I think one must rely on the equations to get something precise. This is why I'm not really a fan of all the delta's you've been using. It's easier to use coordinates because then you can put those coordinates into equations.

Let point 1 lie at ##(c t_{(1)}, x_{(1)})##. This is in the ##S## frame. Equivalently, it also sits at ##(ct_{(1)}', x_{(1)}')## in the ##S'## frame.

Your ##\Delta t' = t_{(1')}##, the time coordinate of ##1'## in the ##S## frame.

Your ##\Delta t'' = t_{(1)}'##, the time coordinate of point ##1## in the ##S'## frame.

Hmm - there's a problem here already, as the inertial observer experiences the longest proper time between events,
Doesn't observer in frame ##x,ct## experiences the shortest time in my case? I would allso like to clear something up before i start writing. Are you writing this for the "passive transformation" or the "active transformation"?

The first line that you are missing is the set of points that are simultaneous with event 1 in the (t,x) frame. Hint: this set of points is parallel to the x axis.

The second line is the set of points that are simultaneous with event 1 in the (t', x') frame. Hint: this set of points is parallel to the x' axis. The important thing to note is that this set of points is DIFFERENT that the set of points of our first line. Otherwise, we won't be using this line.

The third line that you are missing is the set of points that are simultaneous with event 1' in the (t', x') frame. Hint: this is parallel to the x' axis, similar to line 2, but not similar to line 1.

So those are the three lines I suggest you add - now come the points, and the meaning of time dilation on your graph.
Are those lines really needed? If not i would prefer not to include them as they make my picture unreadable. If i want to explain it to someone i have to use multiple pictures and not only one. Otherwise it gets messy.

One important pre-requisite, which I'll call PRE. Make sure that the Lorentz interval, also called the proper time, between the origin and point 1 is the same as the Lorentz interval between the origin and point 1'. I think you addressed this by saying that points 1 and 1' should both be on the same hyperbola of (ct)^2 - x^2 = constant.
How do you mean the same? Explain please.

In the (t,x) frame, line 1 represents the set of points simultaneous with the event you labelled as "1". Find point 2, where this line intersects Ranja's worldline.
I think that my "point 2" is in the origin of frames.

Time dilation says that point if point 1' occurs after point 2, because using the simultaneity conventions of the (t,x) frame, the clock in the (t', x') frame is ticking slowly. (Be sure that the precondition PRE is met!)
"Point 2" is the origin and it is an event which happens before event labeled ##1'## does.

In the (t', x') frame, line 3 represents the set of points simultaneous with the event you labelled as 1'. Find point 3, where this line intersects Ziga's worldline. Time dilation says that point 1 comes after point 3. Again, be sure the precondition PRE is met.
Why are these lines and points so important for me to include in the picture?

How do you mean the same? Explain please.
Prevect is saying that you should ensure the spacetime interval between point 1 and the origin has the same numerical value as the spacetime interval between point 1' and the origin. Saying that they both lie on the same hyperbola (which is a curve of constant spacetime interval) is enough to accomplish this, and I believe that's what you did.

A hyperbola is a curve of constant spacetime interval just as a circle is a curve of constant Euclidean interval (distance).

##\Delta t''## is the time coordinate of the point ##1## with respect to the ##S'## frame.
I thought so.

Your ##\Delta t'' = t_{(1)}'##, the time coordinate of point ##1## in the ##S'## frame.
I could use this to prove that for observer in primed frame thinks time interval of an observer in unprimed frame is longer. I think.

Fredrik
Staff Emeritus
Gold Member
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.

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pervect
Staff Emeritus
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.

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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 paralell 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.

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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.

View attachment 52325

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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.

View attachment 52325
Could you please mark axis in your picture? Picture would make more sense to me then. :)