Is Lorentz contraction explained by space time diagram ?

[sweet springs]

Is there a way to explain Lorentz contraction on space-time diagrams ?
I cannot find a way by myself. Your teaching will be appreciated.

 

[PeroK]

Is there a way to explain Lorentz contraction on space-time diagrams ?
I cannot find a way by myself. Your teaching will be appreciated.

I typed "length contraction spacetime diagram" into Google and got this just for starters:

http://www.phys.vt.edu/~takeuchi/relativity/notes/section13.html

 

[sweet springs]

Thanks.

In your drawing you set $$v=\frac{c}{2}$$.

I read
$$L'=2.6*\sqrt{5}= 5.81 cm$$
$$L=4 cm$$
Thus the formula in the drawing gives $$v=0.73c$$ ?

I am not sure of my reading. I should appreciate your teaching.

 

[PeroK]

Thanks.

In your drawing you set $$v=\frac{c}{2}$$.

I read
$$L'=2.6*\sqrt{5}= 5.81 cm$$
$$L=4 cm$$
Thus the formula in the drawing gives $$v=0.73c$$ ?

I am not sure of my reading. I should appreciate your teaching.

I'm not sure where you are getting those numbers from.

 

[pervect]

Is there a way to explain Lorentz contraction on space-time diagrams ?
I cannot find a way by myself. Your teaching will be appreciated.

It's not quite what you asked for, but Robphy's Physics forum insight article (or his pubished paper). The PF article is "Spacetime Diagrams of Light Clocks" is at is at https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/. While a light clock isn't usually presented as leading to Lorentz contraction, it can do so.

A draft version of the pubished paper, "Relativity on Rotated Graph Paper", is on arxiv. The full version, http://aapt.scitation.org/doi/10.1119/1.4943251 is unfortunately paywalled (last I herad, at least).

The part that may not be explained is how to get length contraction out of the light clock diagram. But the light clock has a constant proper length. The ends of the diamond that represent the light clock are not simultaneous, but if you draw the usual light clock using the rotated graph paper technique (or another technique, but the rotated graph paper technique is especially simple), you can find the projection of the "titled" light clock onto the spatial part of the space-time diagram in a fairly straightforwards manner.

Or you can use multiple light clock diagrams. I'm fairly sure there are some PF posts that go through this missing detail by Rob, but I'm not sure where.

 

[A.T.]

Is there a way to explain Lorentz contraction on space-time diagrams ?

The others provided good links on the standard Minkowski space-time diagrams.

Additionally it might be useful to have a look a space-propertime diagrams, which show the geometric relation between movement, length contraction and time dilation more directly:
http://www.adamtoons.de/physics/relativity.swf

 

[sweet springs]

I'm not sure where you are getting those numbers from.

I just tried to count ruled lines for L and L' in your graph paper drawing http://www.phys.vt.edu/~takeuchi/relativity/notes/section13.html. Sqrt 5 comes from rectangle trigon of length 1,2, and sqrt 5.
Best.

 

[sweet springs]

Thanks perverct and A.T. I expect more obvious showing like the ratio of such and such length in spacetime diagram is gamma but it seems too much.

 

[Mister T]

Thanks.

In your drawing you set $$v=\frac{c}{2}$$.

I read
$$L'=2.6*\sqrt{5}= 5.81 cm$$
$$L=4 cm$$
Thus the formula in the drawing gives $$v=0.73c$$ ?

At a speed of $0.5\ c$ and $L' \approx 5.8$ we should have $L\approx5.0$.

In the figure they've mislabeled either the length $L$ or the length $L'$.

I'd need more context to figure out which.

 

[robphy]

Here is a diagram of length contraction taken from my paper (referenced by @pervect). (Thanks @pervect .)
It's a standard minkowski diagram with worldlines supplemented with light clocks (in accordance with Minkowski spacetime geometry... so that you can measure by simply counting ticks [light-clock diamonds].)

 

[sweet springs]

Thanks robphy. Your light-clock diamonds seem to rely on light travel. Lorentz contraction works between two events of space-distance where no light can travel between. Doesn't it matter?

 

[robphy]

The details of the explanation below are in my paper
and on the https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/ Insight.

The light-clock diamonds are modeled after one tick of an observer's clock.
Once that tick is established along that observer's worldline, that tick becomes the prototype for all measurements (temporal and spatial) made by that observer.
(It's like saying that once I have one little box on a page, I can tile the page with that box to obtain a sheet of graph paper... which of course exploits the symmetries of the plane with a euclidean metric.)

From the graphics in my Insight,...

and then for Bob
(where the key construction is that Bob's light-clock diamonds have the same area as Alice's)

You can tile the plane with Bob's light-clock diamonds.
In my earlier length-contraction diagram, you see Bob's diamonds along his t- and x-axes.You can play with idea here
https://www.geogebra.org/m/HYD7hB9v#material/VrQgQq9R ,
where you can draw line-segments (timelike, spacelike, or lightlike) on a spacetime diagram with the special tool on its toolbar.
The diamonds help you measure the square-interval of the segment using that observer's coordinates.

 

[vanhees71]

In terms of conventional Minkowski diagrams you also find it in my SRT PF FAQ:
http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf