A light cone question for Mordred

[studiot]

Do you have an images of what a light cone of a particle B might look like, when measured in the frame of another particle, A such that A and B are in substantial relative motion?

[Mordred]

I read the thread that led to this question. Give me a bit to find a decent example with an explanatory. Most of the examples I have are in textbooks.

 

http://www.scienceforums.net/topic/100658-simultaneity-split-from-is-time-real/#entry955873

 

Just to clarify your looking for one that specifically shows the following ?

 

Every point on the world line of a particle generates a new light cone.

So considering several observers can generate a complicated tangle of light cones.

It is important not to unintentionally include an assumption of some sort of absolute spacetime in consideration of these.

 

A good question to consider here would be:

 

How do the light cones from particles outside the light cone of some given particle overlap with the light cone of the given particle?

Edited November 13, 2016 by Mordred

[studiot]

Yes this was the motivation for the question.

 

Here is some more background.

 

With reference to the diagram, consider the instantaneous coordinate frame of A with one spatial and one transformed time axis ct (I have kept it simple).

 

The part light cone represented by the triangle EAC contains all the points that particle A could concievably attain and a sample worldline is plotted as a dashed line.

This worldline must lie with EAC.

 

Now consider observers situated at B, C and D, all moving relative to A.

 

Each will have its own lightcone represented by a triangle in its own frame.

 

But each will see time as dilated and length as contracted along the ct and x axes of A.

 

So each will see the figure A sees as triangle AEC and the worldline as distorted compared to the view from A.

 

Hence the question, as it will take me some time to plot it all out.

 

Edited November 13, 2016 by studiot

[Mordred]

This will help save some time while I look for better coverage.

 

http://www.as.utexas.edu/astronomy/education/spring06/komatsu/secure/lecture10.pdf

 

essentially light-cone causality as per the images here

 

http://plato.stanford.edu/entries/spacetime-singularities/lightcone.html

 

This one might be a little too advanced.

 

http://www.rpi.edu/dept/phys/Courses/Astronomy/CurvedSpacetimeAJP.pdf

 

Try Kruskal diagrams

 

 

 

If what I understand your after is accurate. Each event coordinate has its own causal connection. This is represented by seperate lightcones at each coordinate.

 

In flat space the lightcones has an even distribution and same orientation.

In curved space the lightcone distribution becomes more dense as per the individual event coordinates and will change alignment. The first image would be improved by showing the proper alignments. However think of tilted hourglasses.

 

One can find decent coverage of this typically in Schwartzchild metric articles that go into the more advanced coordinate systems. Though I find the Penrose diagrams rather tricky to explain.

 

By the equivalence principle these diagrams are equivalent for inertial observers and gravitational observers.

 

The image here on this site shows a better representation of the lightcone orientation.

 

https://en.m.wikipedia.org/wiki/Light_cone

Edited November 13, 2016 by Mordred

[studiot]

Thanks Mordred, I will study them tomorrow, it's bedtime here right now.

[studiot]

One thing all the examples have in common is that the cones are referred to the same frame axes in any one diagram.

 

What I am interested in is what those cones would look like if redrawn to a different set of frame axes.

 

One thing I have realised whilst considering this is that there is only one direction relative velocity can take with only an x and a ct axis.

[Mordred]

yes the Lorentz transforms only affect the x axis (assumed direction of motion) and the time axis ct. So we graph the x and axis and use trig for the rotations. For example on the last diagram. It shows the turnaround stage on the hyperbola.

For calibrating two observers S and S' we shift the axis themselves. Axis rotation

You can find the procedure here. Notice the shift in the ct' axis on the figure 3.

http://www.google.ca/url?sa=t&source=web&cd=2&ved=0ahUKEwjby5jS36jQAhXC0FQKHaZKBQ0QFggcMAE&url=https%3A%2F%2Farxiv.org%2Fpdf%2Fphysics%2F0703002&usg=AFQjCNH0WYF7gsM-vW-D0_w7iNx2VXyj3A


This is probably closer to what your after with two inertial observers.

Yes this was the motivation for the question.

Here is some more background.

With reference to the diagram, consider the instantaneous coordinate frame of A with one spatial and one transformed time axis ct (I have kept it simple).

The part light cone represented by the triangle EAC contains all the points that particle A could concievably attain and a sample worldline is plotted as a dashed line.
This worldline must lie with EAC.

Now consider observers situated at B, C and D, all moving relative to A.

Each will have its own lightcone represented by a triangle in its own frame.

But each will see time as dilated and length as contracted along the ct and x axes of A.

So each will see the figure A sees as triangle AEC and the worldline as distorted compared to the view from A.

Hence the question, as it will take me some time to plot it all out.

lcone2.jpg

The link above will show how to use two observers in the spacetime diagram on calibrating the two observers to the type of daigram above.

See the "calibration Hyperbola" figure 3 here.



from this site.

http://www.scholarpedia.org/article/Special_relativity%3a_kinematics#Graphic_representation_of_the_standard_Lorentz_transformation

Edited November 14, 2016 by Mordred

[studiot]

Unfortunately the link somehow doesn't work for me.

 

 

 

Edit

 

Stet

 

Got it now.

Thank you for all these links, I am aware of the bones of what they contain but they do offer some fresh views on the material.

 

But they still suffer from a basic issue, from what I can see.

 

I have deliberately restricted the system to one spacial axis and one forbidden rotation scaled time axis to try to preserve simplicity.

 

So far as I can see this requires that all alternative frames of reference must be parallel to these axes.

 

This is the basis of my second comment in post# 6

 

All the examples show an x' axis rotated with respect to the x axis.

 

This is only possible in a system with 2 or 3 (or more) spatial axes.

Edited November 14, 2016 by studiot