# Time dilation in minkowski diagrams

#### center o bass

Hey! I'm trying to understand time dilation in terms of minkowski diagrams.
Below I've added a diagram showing the two coordinate systems where the primed one
moves relative to the unprimed one with a speed v.

My reasoning in this diagram is that when the observer in the unprimed system compares
the clock in the primed system with his own that corresponds to comparing the space-time coordinates with eachother. He therefore assigns (t,x) while the primed observed assigns (t',0). Or in other words, as he sees t on his clock he simultaneously sees t' on the primed observers clock. But from my diagram it clearly seams like t' > t which is not consitent with the result that t = (gamma)t' > t; i.e the unprimed observer should see the moving clock run slow not vice versa.

What is wrong with my reasoning here?

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#### robphy

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Gold Member
In your spacetime diagram, " t' > t " is not correct because you must
measure intervals (spacetime separations) with Minkowskian spacetime geometry---not Euclidean geometry.
In Minkowski spacetime, " t' < t ".

[Note that in a PHY 101 position-vs-time diagram
(which has an underlying [and underappreciated] Galilean spacetime geometry)
we have " t' = t ".]

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#### robphy

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Gold Member
Although the spacetime diagram is drawn on a plane (like a familiar Euclidean geometry diagram is), the problem is how to measure "lengths" along worldlines (physically representing durations of elapsed time).

One way is to draw hyperbolas (for Minkowski spacetime) instead of circles (for Euclidean geometry). These represent the locus of points [events] that are "equidistant" from a given point [event] in the corresponding geometry.

Another way is to trace out light-rays from light-clocks [as done in my avatar].
What is interesting about that diagram in my avatar
is that the spacetime-volume enclosed between each tick
[the volume contained inside
the intersection of the future light cone of one tick and
the past light cone of the next tick] is Lorentz invariant.

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#### center o bass

So could I then conclude that any kind of analysis in such a diagram which really involves lengths or differences in coordinates is invalid? Is for example a similar analysis on length contraction also invalid?

This in contrast to analysis only which involves separate space-time events like this one:

Is that kind of analysis valid because it does not involve distance?

Btw: Could you recommend any material where I could read up on the properties of Minkowski space you're takling about? :)

#### robphy

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Gold Member
has scales drawn along each line...
and that those scales are not consistent with Euclidean geometry in the plane.
In other words, along lines through the origin, the points at "1 tick mark from the origin"
do not trace out a circle... but they do trace out a hyperbola.

t'=0.8 and t=1.0... thus t' < t.

#### bapowell

What is wrong with my reasoning here?
You need to project t' onto the t axis in order to determine how the (x,t) observer views the (x',t') observer's clock. This projection should be taken along the line of simultaneity in the (x',t') observer's frame: draw a line through the event that's parallel to the x' axis. Read off where this line intersects the t-axis.

#### center o bass

You need to project t' onto the t axis in order to determine how the (x,t) observer views the (x',t') observer's clock. This projection should be taken along the line of simultaneity in the (x',t') observer's frame: draw a line through the event that's parallel to the x' axis. Read off where this line intersects the t-axis.
Nah? The (x,t) observer must surely compare the two clocks simulatneously occording to him and not to the other observer?

robphy said:
t'=0.8 and t=1.0... thus t' < t.
Yes, this starts to make sence. Thank you! How do one arrive at the conclusion to draw hyperbolas relating the points rather than circles? I do know that there are several things in minkowski space that look 'hyperbolic', but could you recommend any good texts or derivations?

#### bapowell

Nah? The (x,t) observer must surely compare the two clocks simulatneously occording to him and not to the other observer?
Oops. Got spun around!

#### robphy

Homework Helper
Gold Member
Yes, this starts to make sence. Thank you! How do one arrive at the conclusion to draw hyperbolas relating the points rather than circles? I do know that there are several things in minkowski space that look 'hyperbolic', but could you recommend any good texts or derivations?
Check out
Dray: http://www.physics.oregonstate.edu/coursewikis/GSR/
Moore: https://www.amazon.com/dp/0072397144/?tag=pfamazon01-20 or https://www.amazon.com/dp/0070430276/?tag=pfamazon01-20
Ellis & Williams: https://www.amazon.com/dp/0198511698/?tag=pfamazon01-20

I've got something brewing... but it's not quite ready yet.

#### bobc2

Here are three different frames showing relationship to the hyperbolic calibration curves. I'm showing the event labled "NOW" for each observer such that each observer is located on the same hyperbolic curve (same proper time). Notice that the simlultaneous X1 space for each observer is a line tangent to the hyperbolic curve. So, you can see how the projections from frame to frame give the time dilations.

#### center o bass

I've got something brewing... but it's not quite ready yet.
Nice. Tell me when you do. I liked the notes by moore. Will be reading up on hyberbolic geometry :)

#### bobc2

I think it's easier to see the time dilations using a symetric space-time diagram. Have two observers moving in opposite directions with the same speed relative to a rest frame as shown in the diagram below. Then you can see the derivation of the hyperbolic calibration curves.

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