Understanding Asymptotes in Rindler Space - Physics Forums

[DiracPool]

If you look at this video, Lenny Susskind says that you can look at the null-lines, or light cone, lines of the Minkowski diagram at being equivalent to a zero radius in a uniformly accelerated reference frame. This was surprising to me since it looked as though the lines on the graph clearly showed a traversal of space and time that looks on the graph to be non-zero at these asymptotes. However, it does make sense when you consider that the Lorentz contraction pushes a radius to zero once we hit the extreme of these asymptotes.

My question is does this apply to the time axis also? I mean, once you shrink your "time radius," if you will, down to zero, do the time-like asymptotes also qualify as zero all along their extents as they do for their space-like counterparts? It would seem as if they would. But Lenny doesn't state this explicitly.

Forward to 124:50



Lastly, what are the implication here for a post I made earlier. This post asked the question of whether (effectively) time was zero and space was zero along the asymptotes in the Minkowski diagram. From Lenny's dialog, this would indeed be to seem the case.

https://www.physicsforums.com/threads/minkowski-diagram-questions.838131/

 

[Dale]

From the origin, the locus of all events with a fixed spacetime interval $s^2=-t^2+x^2+y^2+z^2$ is a hyperboloid of one sheet if $s^2>0$ or a hyperboloid of two sheets if $s^2<0$. In either case, the hyperboloid thus formed is asymptotic to the light cone formed by $s^2=0$, which is a degenerate hyperboloid.

In the video he is making the analogy between a hyperboloid and a sphere, where a sphere is the locus of all points with a fixed distance $r^2=x^2+y^2+z^2$ from the origin.

 

[DiracPool]

In the video he is making the analogy between a hyperboloid and a sphere, where a sphere is the locus of all points with a fixed distance r2=x2+y2+z2r^2=x^2+y^2+z^2 from the origin.

He is not specifically talking about a "hyperboloid," per se. In fact, he pointedly voids this distinction by eschewing the participation of the y and z axis, if you look earlier in the video. He's specifically modeling hyperbolas.

The Rindler space model he presents is one dimension of time and one dimension of space. The rest of your comment doesn't address my question, in all due respect, DaleSpam

 

[Dale]

He is not specifically talking about a "hyperboloid," per se.

You are right. I have watched that video, but was going from memory.

Replace all of the 4D comments with corresponding 2D comments and the point remains. The light cone is a degenerate hyperbola which is asymptotic to both timelike and spacelike hyperbolas.

 

[DiracPool]

The light cone is a degenerate hyperbola which is asymptotic to both timelike and spacelike hyperbolas.

Ok, but to address my initial question, can we regard the null-lines as effectively zero radius and zero time?

Let's say notwithstanding the argument that a photon cannot claim to have it's own inertial reference frame. Just from a broad cursory look at what the Minkowski diagram is saying or representing...

 

[DiracPool]

The light cone is a degenerate hyperbola which is asymptotic to both timelike and spacelike hyperbolas.

OK, I re-read that post and am wondering what "asymptotic to both timelike and spacelike hyperbolas" means. Remember, I have a big B as the prefix in the title description thread

 

[Dale]

Ok, but to address my initial question, can we regard the null-lines as effectively zero radius and zero time?

No. This is one difference between a hyperboloid and a sphere. A degenerate sphere is a single point with no extension along any of the axes. A degenerate hyperboloid is a cone with infinite extension along all axes. Similarly with 2D figures.

 

[Dale]

OK, I re-read that post and am wondering what "asymptotic to both timelike and spacelike hyperbolas" means. Remember, I have a big B as the prefix in the title description thread

Sorry. I don't know how to explain it simpler than with reference to the equations for geometric figures. Do you remember the equation of a sphere and of a hyperbola from high school algebra? It may have been a while, but I bet you have seen this before (maybe not with these words, but with the same math)