Understanding Boosts in Misner Space: Where Do They Come From?

[space-time]

I've been trying to look into studying Misner space due to its simplicity (while still containing CTCs). Upon reading this wiki page, I have come across some things that I am just not familiar with.

Here is the wiki page: https://en.wikipedia.org/wiki/Misner_space

Now this is going to sound like a noob question, but what exactly are these boosts that the page speaks of when it says:

(t, x) --> ( tcosh(pi) + xsinh(pi), xcosh(pi) + tsinh(pi) )

Exactly what does the page mean when it says that every pair of spacetime points can be identified by this boost? Also, where does this boost come from? How was this boost decided to exist or be true? This doesn't seem like any of the Lorentz transformations that I know of from special relativity, and it doesn't seem like something that would be derived from the Einstein field equations themselves.

Thanks for any assistance.

 

[PeterDonis]

what exactly are these boosts

The term "boost" comes from the fact that there is an isometry of Minkowski spacetime called the "boost" isometry whose integral curves, in the standard $(t,x)$ inertial coordinates, look like $(t \cosh \alpha + x \sinh⁡ \alpha, x \cosh⁡ \alpha + t \sinh \alpha)$, where $\alpha$ ranges from $- \infty$ to $\infty$. (In a standard spacetime diagram these are hyperbolas.) Each value of $\alpha$ can be thought of as the velocity parameter for a Lorentz boost (i.e., Lorentz transformation without any rotation or translation) into a frame in which an observer moving at $v = \tanh \alpha$ in the original frame is at rest; that's where the term "boost" comes from. But here it's just being used to pick out points that are identified (see below).

what does the page mean when it says that every pair of spacetime points can be identified by this boost

It means that the points labeled by the coordinates $(t,x)$ and $(t \cosh \pi + x \sinh⁡ \pi, x \cosh⁡ \pi + t \sinh \pi)$ are the same point. A simpler example of the same thing is to construct a cylinder from a flat plane by identifying points $(x,y)$ and $(x+C,y)$, where $C$ is the circumference of the cylinder.

where does this boost come from? How was this boost decided to exist or be true?

It doesn't have to "come from" anywhere or be "decided to exist". It's just a mathematical construction. Mathematical constructions can be done however you like as long as they are mathematically consistent.

 

[Martin Scholtz]

I will answer in detail but do you need to clarify the boosts first in general, or just in the example of the Misner space?

Boosts are analogous to rotations but ty they represent transformations between two inertial frames, rather than two frames rotated with respect to each other with zero mutual velocities. The boosts are standard Lorentz transformations although it's perhaps not obvious when written in terms of hyperbolic functions.

If you need to explain this part, I can.

Or do you have difficulties just with understanding the role of boosts in the Misner space? Misner space is locally the same thing as the Minkowski Space but with different global topology which is generated by the identification of different points. Misner space is an example of non trivial space-time although locally it's indistinguishable from Minkowski, because it contains the NUT charge, a topological defect of very peculiar nature. If you need, I can elaborate on this topic as well.

 

[George Jones]

A question related to what @Martin Scholtz wrote: In special relativity, are the you familiar with concept of rapidity, also known sometime as "velocity parameter"?

 

[Martin Scholtz]

A question related to what @Martin Scholtz wrote: In special relativity, are the you familiar with concept of rapidity, also known sometime as "velocity parameter"?

As far as I know (remember), rapidity is just the ratio
$$\beta = \dfrac{v}{c}$$
i.e the velocity expressed in the units of speed of light on vacuum. For example$$\beta=0.5$$ is the half of the speed of light, like 0.5Mach is the half of the speed of the sound in the air.

If you write expressions like $$\sqrt{1-v^2/c^2}$$, in terms of rapidity it would be simply $$\sqrt{1-\beta^2}$$.

However, in theoretical physics it is customary to choose geometrical units in which the speed of light is dimensionless and equal to $$c=1$$. Hence, the rapidity is the same thing as velocity and instead of replacing $$v/c$$ by $$\beta$$ we simply omit $$c$$ from all equations. Value $$v=0$$ corresponds to rest, value $$v=1$$ corresponds to the motion with the speed of light.

 

[PeterDonis]

rapidity is just the ratio

No, that's just velocity in natural units. Rapidity is $\alpha$ in what I posted earlier; the velocity $v$ in natural units, what you call $\beta$, is given by $\beta = \tanh \alpha$.

 

[Martin Scholtz]

Yes, of course, you are right. I don't use this terminology so I remembered it wrong. Thanks for the correction.