time dilation in Minkowski diagram

[michel123456]

In the Minkowski diagram article in wikipedia, under the paragraph Time dilation, here below the following graph:

 

 

 

Quoted from wiki:

Time dilation: Both observers consider the clock of the other as running slower.

 

and the beginning of the explanation:

Relativistic time dilation means that a clock moving relative to an observer is running slower and finally also the time itself in this system. This can be read immediately from the adjoining Minkowski diagram. The observer at A is assumed to move from the origin O towards A and the clock from O to B. For this observer at A all events happening simultaneously in this moment are located on a straight line parallel to its path axis passing A and B.

 

Emphasis mine.

 

My question is the following: Can A observe B? I think not, because light travels at C, which is a diagonal not represented on this graph.

 

IMHO what A is observing is the point B2 in the following graph:

 

 

Point B is simultanate but it is not observable.

Edited February 23, 2011 by michel123456

[Schrödinger's hat]

Point B is simultanate but it is not observable.

 

Good observation. You're right about the interval between all those events being spacelike.

This doesn't really change the point of the diagram though. Both observers could carefully record everything they did, then observe what they saw of the other some time later. They could then determine that, according to their reference frame (what is real for them) is that the other clock is running slower.

There would be lots of factors to take into account before they could reach this conclusion from their measurements.

 

One thing that trips people up is: if you are moving away from one another, then everything red shifts due to doppler effect. As the light you see redshifts, so too do the events it records. Say the clock was a beeping radio. The beeping you receive would be even slower than the rate at which the radio beeps in your reference frame, because each beep would have to travel further.

It's important to note that this is not the time dilation, it's simple, non-relativistic doppler shift, and could be reproduced by an audible signal of an object moving near the speed of sound. If they turned around, the beeps would speed up.

 

Once you correct for this and look at all the measurements you made, you still find the beeps are slower. Both parties will see that the other has the slow clock. This is time dilation.

It's not important that they agree on the times and positions at which things happen, as they are not in the same frame.

[michel123456]

Thank you for the reply.

I can't recall exactly what was the next step. I always have a next step...

 

What I can say now that I look again at the wiki entry, is that it is drawn in a way that people could believe point A actually looks at point B, and point B looks to C. That is not the case.

Edited February 28, 2011 by michel123456

[Sisyphus]

Which reference frame are the yellow lines the speed of light for? It looks like a third frame.

[Schrödinger's hat]

Which reference frame are the yellow lines the speed of light for? It looks like a third frame.

 

Uuhm, look closer. Note how the x,ct and x',ct' don't make 90 degrees? Everything here is drawn in a frame such that the yellow lines are the speed of light, and Alice and Bob are travelling at equal velocities in opposite directions.

 

This brings me to something I've been musing over for a while. I sort of re-invented Minkowski diagrams without knowing they existed a while back, but adding a few features and musing for a while lead me to some geometric techniques whereby some reasonably advanced SR student problems (Anything involving instant acceleration, and all of the 'paradoxes' that undergrads are given to muse over) can be solved without any calculators or algebra.

 

This lead to an idea for an ebook or set of notes for learning SR without an algebra requirement.

 

Does anyone think this is a good idea?

 

Edit: Plain forgot to answer michel before I went off on my tangent. You could alter the diagram so that the lines represented lightspeed signals, I suppose, but that would distract from the issue at hand and bring up all those issues about signal propagation delay and such.

I think it is simplest to just have a line drawn parallel to the x axis in each frame, this is a line of events saying "The times is ___" in the given frame.

It could be made clearer that this is what these are, by extending the line, removing the arrow head and putting crosses where it intersects with each world line.

 

Looking into the way the wiki page is structured, they first take the geometry as a postulate, and hold off on talking about the constancy of lightspeed (these conditions are equivalent) until later.

This is not a good pedagogical method for SR in my opinion, but then this is not the role of an encyclopaedia. It could possibly be structured in a way that was better for teaching without compromising its role as a reference, but going through a derivation SR for every tool or representation introduces a lot of redundancy.

Edited March 1, 2011 by Schrödinger's hat

[michel123456]

Which reference frame are the yellow lines the speed of light for? It looks like a third frame.

 

This is not my invention, You can find the yellow lines as bisectors in the wiki article under constancy of the speed of light.