Singularities and Rindler horizon

[zonde]

I am trying to understand things around singularities and related to this I have a question.

What kind of singularity is Rindler horizon?
Wikipedia (Rindler coordinates) says that: "The Rindler coordinate chart has a coordinate singularity at x = 0,"
But if Rindler coordinates are not defined on the other side of the horizon then they are not related by bijection to inertial coordinates. And then Rindler coordinates do not cover region around x=0 from the other side (I believe this is requirement to state that x=0 is coordinate singularity).

 

[PeterDonis]

What kind of singularity is Rindler horizon?

A coordinate singularity, just like Wikipedia says.

But if Rindler coordinates are not defined on the other side of the horizon then they are not related by bijection to inertial coordinates. And then Rindler coordinates do not cover region around x=0 from the other side (I believe this is requirement to state that x=0 is coordinate singularity).

No, it isn't. Defining a coordinate singularity can be done without any information at all about what other charts the Rindler chart has a bijection with, or even what other points besides x = 0 the chart covers. The reason x = 0 is a coordinate singularity, as the Wikipedia page says, is simply that the metric tensor has vanishing determinant there. That's all. And this fact can be calculated purely from the metric coefficients at x = 0, without any other information required.

 

[TrickyDicky]

I really think the term "coordinate singularity" is an unfortunate one for what it describes, making it easy for people to confuse it with singularities.
All it is needed to know wrt to coordinates and general manifolds is that they cannot generally covered with only one chart of coordinates and therefore there will be points in the manifold that will require a different chart than the one we are using.
The fact that the most frequently used space (euclidean) can be covered with just one chart is a special case that can lead to confusion.
But even with flat manifolds, if we choose to use curvilinear coordinates, the issue shows up, like is the case with Rindler coordinates in Minkowski space.
This really has nothing to do with singularities, thus I think the term "coordinate singularity is misleading.

 

[DrGreg]

Note that Rindler coordinates are defined only for x > 0, so strictly speaking it doesn't even make sense to refer to x=0 as that is an undefined concept. Nevertheless it's fairly standard terminology to say that there's a coordinate singularity "at x=0" on the understanding that this a shorthand for a more precise statement about behaviour in the limit as x→0.

 

[grav-universe]

The Rindler horizon is not really a "singularity" per say. As viewed from the perspective of a stationary inertial frame, it is the distance behind an accelerating observer beyond which a photon will never catch up to the observer that is accelerating from rest with a constant proper acceleration. It is not just a result from SR, but is the same for Galilean kinematics. If a bus starts to accelerate away from a curb and a late-comer runs toward it at a constant rate behind it, when starting out beyond a certain distance behind it, the runner will never catch up. It is the same thing for a photon with relativistic acceleration. The only difference is that one might think that the photon should eventually catch up since the speed of the photon is always greater than the instantaneous speed of the accelerating observer, but that is not the case. Working out the time to catch up to a bus from beyond a particular distance will give a complex (imaginary) result, but that does not mean that those coordinates do not exist, any more than that distance would be considered a singularity. The complex result only means that the runner doesn't catch up. The same goes for light and the Rindler horizon.

 

[zonde]

No, it isn't. Defining a coordinate singularity can be done without any information at all about what other charts the Rindler chart has a bijection with,

Yes

or even what other points besides x = 0 the chart covers.

As I understand it, this is not correct.
Chart has to cover neighbourhood of singularity. So if chart does not cover x<0 then x=0 is not a singularity.

Where are you looking for definition of singularity? I was looking in wikipedia Mathematical singularity
It says for example: "In real analysis singularities are also called discontinuities."

 

[zonde]

Note that Rindler coordinates are defined only for x > 0, so strictly speaking it doesn't even make sense to refer to x=0 as that is an undefined concept.

x=0 is well defined concept. It's Rindler coordinates that are not defined for x=0.

Nevertheless it's fairly standard terminology to say that there's a coordinate singularity "at x=0" on the understanding that this a shorthand for a more precise statement about behaviour in the limit as x→0.

What is key requirements that a point has to fulfil to be considered a singularity?

 

[zonde]

The Rindler horizon is not really a "singularity" per say. As viewed from the perspective of a stationary inertial frame, it is the distance behind an accelerating observer beyond which a photon will never catch up to the observer that is accelerating from rest with a constant proper acceleration. It is not just a result from SR, but is the same for Galilean kinematics. If a bus starts to accelerate away from a curb and a late-comer runs toward it at a constant rate behind it, when starting out beyond a certain distance behind it, the runner will never catch up. It is the same thing for a photon with relativistic acceleration. The only difference is that one might think that the photon should eventually catch up since the speed of the photon is always greater than the instantaneous speed of the accelerating observer, but that is not the case. Working out the time to catch up to a bus from beyond a particular distance will give a complex (imaginary) result, but that does not mean that those coordinates do not exist, any more than that distance would be considered a singularity. The complex result only means that the runner doesn't catch up. The same goes for light and the Rindler horizon.

If you view it as a physical question then sure there are no such a thing as physical singularity. So the question is mathematical.

 

[DrGreg]

x=0 is well defined concept. It's Rindler coordinates that are not defined for x=0.

Well, on a technicality, no. If the definition of Rindler coordinates says the definition applies only for x > 0, then x = 0 really is undefined, even though you may have an equation in which it is possible to put x = 0 and get a unique answer. It's already been stipulated that you're not allowed to do that. And why not? Look at the transformation in the other direction [itex]x = \sqrt{X^2 - T^2}[/itex]. For x=0, this doesn't just have the single solution X=0, T=0, it has multiple solutions X=±T for all values of T. That's what makes it a coordinate singularity, the fact there isn't a one-one mapping at x=0.

If we restrict attention to the subset of spacetime where x > 0, there is a bijection (one-to-one mapping) between Minkowski coordinates and Rindler coordinates. If you try to extend the subset to include all the events where "x = 0" you no longer have a bijection.


Note: throughout this post I have been using the notation of the Wikipedia article Rindler coordinates.

 

[zonde]

That's what makes it a coordinate singularity, the fact there isn't a one-one mapping at x=0.

Note: throughout this post I have been using the notation of the Wikipedia article Rindler coordinates.

Where do you take conditions of what makes (coordinate) singularity?

 

[TrickyDicky]

Where do you take conditions of what makes (coordinate) singularity?

Zonde you seem to be again confusing chart's limitations in manifolds with singularities despite #3.
They have no factual relation, they are called coordinate singularities but they are not singularities. If you want to talk about singularities that is fine but they have nothing to do with your issue with the Rindler chart.

 

[grav-universe]

If you view it as a physical question then sure there are no such a thing as physical singularity. So the question is mathematical.

Well, it is the physical aspects that provide the necessary insight, while the mathematics describes them. If we define a singularity as a place with infinite acceleration, then that would lie at a point mass singularity with Newtonian gravity, for example, the place where the acceleration of gravity would become infinite. In GR, it would lie at the event horizon at r = 2m with that definition, although some might still consider the point at r = 0 to be the singularity, depending upon whether one defines a singularity as infinite acceleration or the central gravitational point that all mass must fall to internally, I would say the former, although the coordinate system can also be changed from that of Schwarzschild so that the event horizon can be shrunken to a point and the interior coordinates are no longer accounted for, leaving only the exterior coordinate system that external observers can physically measure with a singular point mass at the center, which I personally prefer.

So with Rindler, going on the definition of infinite acceleration, if a photon beyond a certain distance cannot ever catch up to an observer with a finite constant proper acceleration, then a massive particle at that distance with a speed of less than c will certainly never catch up either. At x > 0, a photon could catch up within finite time. At x = 0, the photon would not catch up or would just catch up after infinite time. But since the speed right at x = 0 must be c to catch up just at infinite time, and since a massive particle must infinitely accelerate to get to a speed of c, then infinite proper acceleration must be applied to the massive particle at x = 0 to catch up to the accelerating observer, so that point where infinite acceleration must be applied could be considered the "singularity" in a sense.

But while x < 0 can always be plotted from the perspective of an inertial frame, it cannot in the accelerating frame. Let's say we have a ship where the accelerating observer is at the front, with the front accelerating with some finite constant proper acceleration and the back of the ship lies at the Rindler horizon in relation to that proper acceleration at the front. In order for the ship to maintain Born rigidity, so that if all parts of it were to stop accelerating at some point, different parts at different times depending on the inertial frame of observation due to relativity of simultaneity and the different instantaneous speeds at different times along the ship, such that it will have been contracted correctly according to SR upon reaching the final frame and becoming inertial again, then we will have different constant proper accelerations applied all along the length of the ship. But in order for the back of the ship to continue to remain caught up with the front, since the back lies at the Rindler horizon, it must infinitely accelerate.

In other words, the ship must remain intact in order for the ship's coordinate system to also remain intact and applicable. But if the back of the ship lies beyond the Rindler horizon, it will not remain intact, but will break up, since it would require greater than infinite acceleration at points x < 0 to remain caught up with the front. So those points in the ship's coordinate system are no longer applicable, as they will no longer be attached to the rest of the ship, but rather will have detached from the ship and its overall coordinate system. I suppose the difference between a physical singularity and a coordinate singularity, then, would be that with a physical singularity, infinite acceleration acts upon a particle naturally as with gravity, but with a coordinate singularity, infinite acceleration must be applied to the particle in order for the coordinate system to remain intact and include the particle, but does not naturally occur.

 

[zonde]

Zonde you seem to be again confusing chart's limitations in manifolds with singularities despite #3.
They have no factual relation, they are called coordinate singularities but they are not singularities. If you want to talk about singularities that is fine but they have nothing to do with your issue with the Rindler chart.

Well, the problem is that coordinate singularity at the pole of spherical coordinate system seems very closely related to mathematical singularity.

 

[zonde]

Well, it is the physical aspects that provide the necessary insight, while the mathematics describes them. If we define a singularity as a place with infinite acceleration,

No, in this particular context I am not interested in new definitions. I am interested in established definitions.

 

[grav-universe]

No, in this particular context I am not interested in new definitions. I am interested in established definitions.

What I am saying there is that the Rindler horizon could be considered a "singularity" in the sense that an infinite amount of acceleration would have to be applied to a particle at the horizon in order for it to remain within the accelerated coordinate system. But since it is a coordinate effect, what would be required to keep the coordinate system intact at the horizon in order to contain the particle, not a physical effect that occurs naturally, it is only a coordinate singularity.

 

[pervect]

What one needs to work with Rindler coordinates is simple enough, if you drop some of the advanced discussion from Wiki it has the basics - though I'm not quite sure offhand if their formulation of the coordinates is exactly the same as the one I'm used to working with.

http://en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=514563313

One can see from the plot that Rindler coordinates only cover 1/4 of space-time.

One can play around with extending them, but I'd rather not be responsible for any physical confusion one may get as a result. I don't see any obvious way to make them cover more than 1/2 of space-time in any event - they currently cover 1/4.

My position (which isn't my personal invention and I believe is widely accepted) is that coordinates have absolutely no physical significance whatsoever, just as it doesn't matter whether you label a map square "b5" or "g7". This makes it impossible to get physically confused by any coordinates. Unfortunately I've seen a fair number of people at PF who manage to confuse themselves via coordinates. I'm not sure how, exactly, though it would seem likely that they can't possibly share the above philosophy if they do.

My best guess is that the confusion arises at least in part by applying certain shortcuts that really only work in specific types of coordinate systems and assuming they work in general - or even worse, taking the "shortcuts" they are used to using as some sort of basic, fundamental definition.

In any event, this sort of confusion seems to be really hard to cure - I make occasional remarks in the hope it may help...

 

[zonde]

My position (which isn't my personal invention and I believe is widely accepted) is that coordinates have absolutely no physical significance whatsoever, just as it doesn't matter whether you label a map square "b5" or "g7".

Yes, this is rather obvious idea. Physical reality does not change if we change it's description.

This makes it impossible to get physically confused by any coordinates. Unfortunately I've seen a fair number of people at PF who manage to confuse themselves via coordinates. I'm not sure how, exactly, though it would seem likely that they can't possibly share the above philosophy if they do.

Well, I think that there is place for confusion if you take infinity as a valid coordinate or vice versa.

Do you know about Achilles and the tortoise paradox? It is not about coordinate systems but we can pretend like it is. We can attach to each step in this paradox a new coordinate label for time and position.
And the point of this paradox is that Achilles can never overtake the tortoise. Rather physical statement, right?