Symmetrically Opposite Solution to Special Relativity

[wmikewells]

I read some place that there is an symmetrically opposite solution to special relativity where time goes backwards, but darn if I remember where I read it. The opposite solution is largely ignored because it does not have any real world application.

I know that traditional special relativity has three effects on a reference frame moving relative to me:

1. rulers are contracted
2. clocks go slower
3. leading clocks are adjusted back in time and trailing clocks are adjusted ahead in time

I was just curious what effect the opposite solution to special relativity had on rulers and clocks. I played with the rulers and clocks for a bit, but I couldn't come up with the solution.

 

[Fredrik]

I don't know what "opposite" you have come across. It doesn't sound like anything I've heard of, so my first guess would be that you just misunderstood something at the time you read it. I suppose you could be talking about the fact that the "time reversal" operator
\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix} satisfies the definition of "Lorentz transformation".

You really need to have a reference when you ask for an explanation of something you've read, especially when it's something that's this hard to identify.

 

[Dale]

I read some place that there is an symmetrically opposite solution to special relativity where time goes backwards, but darn if I remember where I read it.

I am not certain, but it sounds like you may be talking about the full Lorentz group, O(1,3). The part that usually pertains to special relativity is just the restricted Lorentz group SO+(1,3). Here is a brief overview:
http://en.wikipedia.org/wiki/Lorentz_group#Connected_components

 

[wmikewells]

Thank you for your replies. I give my apologies for not being able to produce the source of the idea. I'll take a look at the restricted Lorentz group link to see if that is what I was referring to.

Unfortunately, I was surfing the Internet while looking into another issue, and I read the paragraph or two about the opposite solution. I thought it was cool, but I was after other fish, so I filed it away in my memory and went on. I am not so sophisticated as to understand what the "time reversal operator" is, but the gist of the idea I happened upon is this (if my memory serves me correctly).

Special relativity is like an orthogonal projection with a slight rotation. It is a common tool in explaining special relativity although I can't seem to find any diagrams in Wikipedia as I am writing this. The rotation accounts for the length contraction and time lengthening. I have seen this explanation in several popularized versions of special relativity. Standard special relativity rotates in one direction. However, the symmetrical opposite solution rotates in the opposite direction. The opposite rotation presents a solution where time goes backward, so it has no real world application. I wish I was more expert in relativity to explain in detail what I was talking about.

I'll try to fill in the missing pieces as I wait for a reply.

 

[wmikewells]

I took a look at the link above (http://en.wikipedia.org/wiki/Lorentz...ted_components ), and unfortunately, it is above my pay grade. I'll try to find an example of the orthogonal rotation example I mentioned, and maybe I'll get lucky and find an explanation about the opposite solution.

 

[Fredrik]

In 1+1 dimensions, a Lorentz transformation is a matrix
$$\frac{\sigma}{\sqrt{1-v^2}}\begin{pmatrix}1 & -v\\ -\rho v & \rho\end{pmatrix}$$ where $v$ is the velocity, and $\sigma,\rho\in\{-1,1\}$. A restricted Lorentz transformation is one with $\sigma=\rho=1$.

A Lorentz transformation with $\sigma=-1$ changes the sign of the time component when it acts on a coordinate pair of an event on the time axis. In this sense, it "reverses the direction of time".

 

[WannabeNewton]

I think he/she may be asking how time dilation and length contraction are affected/reversed, if at all, under the elements in the connected component of $O(3,1)$ which corresponds to non-orthochronous Lorentz transformations.

EDIT: I should add that what you normally think of as time dilation and length contraction are simply special cases of Lorentz boosts applied to the coordinate representations of position vectors. Any Lorentz boost can be decomposed into consecutive infinitesimal transformations belonging to the connected component of $O(3,1)$ which contains the identity transformation (the decomposition will start with the identity transformation). The time reversal Lorentz transformation (which is non-orthochronous by definition) belongs to a connected component which does not contain the identity so I am still unsure about what you are even trying to describe physically.

 

[Fredrik]

Special relativity is like an orthogonal projection with a slight rotation.

It's not. A restricted Lorentz transformation is however a hyperbolic rotation. This can be seen by defining an angle $\theta$ by $\tanh\theta=v$, and rewriting the Lorentz transformation in terms of that angle (the "rapidity") instead of the velocity.

Standard special relativity rotates in one direction. However, the symmetrical opposite solution rotates in the opposite direction. The opposite rotation presents a solution where time goes backward, so it has no real world application.

It could be that what you have seen is that when we look for transformations between inertial coordinate systems that are consistent with the principle of relativity and rotational invariance of space, we find reflections, Galilean boosts, restricted Lorentz transformations (i.e. hyperbolic rotations of spacetime), and plain old rotations (of spacetime). The rotations can be ruled out by other things. One of the problems would be that we would be able to rotate the "time axis" all the way down to what was previously the "space axis", so that what one inertial observer thinks of as velocity 0 is infinite speed to another. These arguments are far more difficult to understand than special relativity. They are also more difficult than the stuff about connected components of the Lorentz group.

 

[Bill_K]

I read some place that there is an symmetrically opposite solution to special relativity where time goes backwards, but darn if I remember where I read it. The opposite solution is largely ignored because it does not have any real world application.

FWIW, I think the OP is talking about Lorentz transformations applied to tachyons.

 

[Fredrik]

FWIW, I think the OP is talking about Lorentz transformations applied to tachyons.

That's possible too. I didn't think of that for some reason.

wmikewells, you should see by now that this is impossible. We can't keep guessing what you might have heard at some point, when all you remember is that it involves time going backwards, and maybe rotations.

 

[wmikewells]

I will try to find the original reference. I thought it might be an obscure, but common piece of knowledge for those in the know, something along the lines of the Terrell–Penrose effect, which I just recently learned about (not that the two are related). Thanks for trying to make sense of my confusion.

 

[wmikewells]

I figured out one half of my mystery. I did a little more digging. The model I was looking for is the loaf of bread model of time. If my frame and another frame (let's say the star Vega) are not moving with respect to each other, our "slice" of time will be perpendicular slice. However, if the other frame is a rotating planet around Vega, the "slice" of time for the Vega planet will be a French cut (diagonal). The French cut will reach into my past if the Vega planet is receding from me, and it will reach into my future if the Vega planet is approaching me. I mistakenly believed that only one French cut was possible and that special relativity presented that one French cut. When I read about a symmetrically opposite Lorentz contraction solution, I wrongly assumed that it represented the other French cut. However, special relativity accounts for both French cuts: one into my past and one into my future. So, one half of my mystery is resolved, but the other half of my mystery is still open. The thing that has thrown me for a loop is that the source for the symmetrical opposite solution was not some cracked pot on physics forum like myself. Maybe one of the books I have read recently. I'll dig a little more and open another thread if I find it.