Time dilation in minkowski diagrams

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Discussion Overview

The discussion revolves around understanding time dilation through Minkowski diagrams, exploring the implications of spacetime geometry on the perception of time between different observers moving relative to each other. Participants analyze the relationships between time coordinates in different frames and the geometric interpretations of these relationships.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions their understanding of time dilation as represented in a Minkowski diagram, noting a perceived inconsistency in the relationship between time coordinates.
  • Another participant asserts that the comparison of time coordinates must adhere to Minkowskian geometry, indicating that "t' < t" is the correct interpretation.
  • There is a suggestion that it might be possible to analyze the problem using Euclidean geometry, which prompts further discussion on the validity of such an approach.
  • Participants discuss the necessity of using hyperbolas to represent spacetime intervals in Minkowski diagrams, contrasting this with circles used in Euclidean geometry.
  • One participant seeks clarification on whether analyses involving lengths or differences in coordinates are invalid, while others provide insights into valid analyses that focus on separate spacetime events.
  • There are repeated mentions of the need to project time coordinates correctly to understand how different observers perceive each other's clocks, emphasizing the importance of simultaneity in different frames.
  • Several participants express interest in further reading on the properties of Minkowski space and hyperbolic geometry, sharing resources and texts for deeper understanding.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using Euclidean geometry for analyzing time dilation, with some asserting that only Minkowskian geometry is valid. There is no consensus on the best approach to visualizing and understanding these concepts, indicating ongoing debate and exploration.

Contextual Notes

Participants highlight the limitations of using Euclidean geometry in the context of spacetime diagrams, noting that the scales and measurements differ fundamentally from those in Minkowski spacetime. The discussion also reflects varying levels of familiarity with hyperbolic geometry and its implications for understanding time dilation.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of physics, particularly those interested in special relativity, spacetime concepts, and the geometric interpretation of physical phenomena.

center o bass
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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.

http://mindseye.no/wp-content/uploads/2012/01/time1.png

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 each other. 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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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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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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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:
https://www.physicsforums.com/showthread.php?t=407012

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? :)
 
Note that diagrams in the thread you linked to
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.

Can you see your diagram in the diagrams in that posted-thread?
t'=0.8 and t=1.0... thus t' < t.
 
center o bass said:
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.
 
bapowell said:
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:
Can you see your diagram in the diagrams in that posted-thread?
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?
 
center o bass said:
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! :cry:
 
  • #10
center o bass said:
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
Taylor & Wheeler: http://www.eftaylor.com/download.html#special_relativity [see part two].
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.
 
  • #11
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.
SpaceTimeGridB2BB.jpg
 
  • #12
robphy said:
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 :)
 
  • #13
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.
ProperTimeCurves3.jpg
 
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