Is Event Sequencing Relative in the Theory of Manifolds?

[Kommandant]

Is event sequencing relative? Lately I have been thinking of a thought experiment involving a proof for relative event sequencing.

Absolute sequencing would involve event "a" triggering event "b" and ANY observer perceiving event "a" happening first and event "b" happening second.

Relative sequencing would involve event "a" triggering event "b" and SOME observers perceiving event "b" happening first even though it was triggered by event "a."

 

[Rasalhague]

In special relativity, which describes flat spacetime (spacetime in the absence of gravity), and posits that causality propagates no faster than c, it's possible to give a coordinate-indepentent sequence to events if and only if there's a big enough time and a small enough space between them for one event to cause the other.

I gather it's an open question whether this is true in general. Some scanarios have been proposed which might lead to what are called "closed timelike curves", which (if I've understood this right) would have no natural global orientation. But these are exotic and purely hypothetical beasts.

 

[Ken Natton]

From the answer you have just posted on another thread Rasalhague, I’m not in any doubt that you are someone with a much better understanding than my own. However, from something I read only quite recently, my understanding is that what Kommandant is proposing is not possible. It is all to do with the fact that the time term in the Minkowski equation is negative. If it is positive, then on a space time diagram with the cause event at the origin, the effect event lies on a circle centred on the origin, and thus raises the possibility of a cause and effect paradox. With the time term negative, the curve produced is a hyperbola. Different observers have different ideas of the spatial distance and time interval between the events, but all observers see the cause event happening first and the effect event happening second.

There is a further subtlety to it, as you suggest. The pair of curves produced with a vertical transverse axis would represent events that were caused by the event at the origin (northern curve) and events that caused the event at the origin (southern curve). The full diagram would have another hyperbola with a horizontal transverse axis that would appear to suggest the possibility of a cause and effect paradox but for the fact, as you mentioned, that they represent events sufficiently distant in space that light cannot travel from the effect event to the observer in less than the time interval between the events. This is another point about travel faster than light, it would re-raise the possibility of a cause and effect paradox.

 

[Passionflower]

In GR there is no guarantee (unless we impose certain energy conditions) that one event happens before another event for all observers. Also the distance between two events is not neccesarily unique.

 

[yossell]

In SR, for any two events A and B that are space-like separated, there are frames relative to which (a) A is earlier than B, (b) A is simultaneous with B; (c) A is later than B.

In SR, for any two events A and B that are time like separated, if A is earlier than B in one frame, it is earlier than B in all frames. Moreover, if A and B are time-like separated, then it is (theoretically) possible for someone at A to, for example, throw a brick at someone at B, and so have some influence at B - i.e. cause things to happen at B.

In GR, all the same things are true locally. However, globally, there exist solutions of GR where things are able to loop through space time into their own past, and, globally, the idea of a past and future break down somewhat. Though such solutions of GR are physically possible, they appear not to describe our universe.

One final thing I'd like to add, which will undoubtedly be seized on and misunderstood: causation and signalling aren't quite the same thing. Some people formulate relativity in terms of the idea that *information* can't travel faster than light - e.g. there had better be no way of *synchronising* distant clocks else relativity is kaput. But this limits the speed of communication rather than the speed of causation. So some think that a certain kind of uncontrollable faster than light causation is still compatible with relativity.

 

[Ken Natton]

Okay guys, again I have to concede superior understanding to you. But it does seem to me to be a serious problem if you allow it. The more usual example that I have encountered, that is perhaps easy to conceive, uses billiard balls. (Would I better say pool balls for you?) In the reference frame of being in the room with the player, it is clear that the player strikes the cue ball, then the cue ball strikes the object ball, then the object ball falls into the pocket. If a cause and effect paradox is possible, certain observers would see the object ball fall into the pocket before the cue ball strikes the object ball, which they see before the player strikes the cue ball. Clearly, that is highly paradoxical. Extending the idea even further, imagine the situation where a car on the motorway (freeway) is brought to a halt by an obstruction, and a queue quickly builds up behind. In certain reference frames, observers would see the cars at the back stopping before the cars at the front had reached the obstruction. If you ponder that further you would realize that that would mean cars perceived as being at the back by the ground observer would be perceived by the moving observer as further forward and the cars that the ground observer believed reached the obstruction first would actually have to magically pass through the rear most cars to reach the obstruction.

 

[Ich]

I think yossell is referring to entanglement. I wouldn't call this "a certain kind of uncontrollable faster than light causation", I'd call it correlation. Extremely weird, but no cause-effect relationship.

 

[yossell]

Ken!

I'm not sure anything I say contradicts what you say. What's the serious problem you're referring to? In the odd solutions for GR, there are closed causal loops - but this isn't the same as relativity of causation, there's not relativity about what causes what. Rather, there are these strange circular chains of causation.

On a more general point, the notion of causation doesn't appear directly in formulations of SR or GR. There's spatial-temporal separation, there are time like and space like curves and so on - but not causation. Even the direction of time, what is past and what is future, doesn't obviously appear within the fundamental equations of relativity themselves, and there are some who try to analyse temporal orientation in terms of the increase of entropy or chaos, say. Accordingly, I think one has to work quite hard to show that backwards causation truly is paradoxical, even in SR and GR.

 

[yossell]

Hi Ich,

my final paragraph was just meant to be purely theoretical. I just wanted to indicate that there are some who think relativity only forbids faster than light signalling, and that this is at least *theoretically compatible* with certain kinds of faster than light causation. I didn't personally want to take a stance on whether entanglement is in fact such a phenomenon - but perhaps, assuming a certain view about the collapse of the wave function, it would be.

 

[bcrowell]

Re Ken Natton's billiard ball example: This is exactly the classical system used by a group at Cal Tech in the 90's to investigate these issues. Echeverria, 1991, "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory, http://authors.library.caltech.edu/6469/ There is a good popular-level presentation of the subject in ch. 14 of Kip Thorne's book Black Holes and Time Warps. The thrust of the CalTech work was that just because a CTC exists, that doesn't necessarily imply that you get time-travel paradoxes. Essentially they tried really hard to produce time-travel paradoxes with a toy classical system, and they couldn't find any.

Although there are solutions to the field equations of GR that have CTCs, that doesn't mean that our universe has ever had or can ever have CTCs. This is what the chronology protection conjecture is about: http://en.wikipedia.org/wiki/Chronology_protection_conjecture

A good popular-level book about this topic is Gott, Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time.

 

[bcrowell]

Another thought: Causality is a very difficult concept to define in general, and it means different things in different contexts (classical versus quantum mechanical, etc.). A related but more straightforward concept is whether a given theory has predictive power. For a classical theory like GR, this is relatively easy to define: given certain types of boundary conditions, does the theory have a unique solution? One thing that makes this easier to define than causality is that you don't have to worry about an arrow of time (which is typically defined by thermodynamics). A serious foundational problem with GR is that it doesn't appear to be a valid predictive theory under all circumstances. It generically has solutions with singularities, and singularities typically make it impossible to make predictions. Also, it has solutions with CTCs, and what the Cal Tech group's work was about was essentially the question of whether GR loses the ability to make predictions in the presence of CTCs (which is a different question than whether causality is violated). Although the Cal Tech group always found at least one valid solution to the billiard ball problem for any set of initial conditions, there were typically more than one, which leads to the question of how nature chooses which one to do.

Basically this kind of thing is the motivation behind the intense interest in the chronology protection conjecture and the cosmic censorship hypothesis. For these two hypotheses to hold is basically the bare minimum that we need if GR is to be a viable, well-founded theory.

 

[starthaus]

In special relativity, which describes flat spacetime (spacetime in the absence of gravity), and posits that causality propagates no faster than c, it's possible to give a coordinate-indepentent sequence to events if and only if there's a big enough time and a small enough space between them for one event to cause the other.

$$dt'=\gamma(dt-vdx/c^2$$

where $$dt$$ is the temporal separation between events and $$dx$$ is the spatial separation between events in frame F. v is the speed of another frame F' wrt F.
$$dt'$$ is the temporal separation between events in frame F'

It is easy to see that there is nothing compelling $$dt'$$ to have the same sign as $$dt$$ unless $$dt/dx>v/c^2$$ or $$dx/dt<c^2/v$$

Now, $$c^2/v>c$$ so, the above reduces to $$dx/dt<c$$ which happens trivially if we accept "no faster than light" signalling.

 

[JesseM]

$$dt'=\gamma(dt-vdx/c^2$$

where $$dt$$ is the temporal separation between events and $$dx$$ is the spatial separation between events in frame F. v is the speed of another frame F' wrt F.
$$dt'$$ is the temporal separation between events in frame F'

It is easy to see that there is nothing compelling $$dt'$$ to have the same sign as $$dt$$ unless $$dt/dx>v/c^2$$ or $$dx/dt<c^2/v$$

That's true for a particular choice of two inertial frames with a relative velocity of v; if two events have coordinate separation dx and dt in the unprimed frame, then the other frame will agree on the order of the events provided dx/dt < c^2/v (note that c^2/v is a faster-than-light speed). However, it's also worth noting that in any case where |dx/dt| > c, then it will always be possible to find some other frame with a velocity of v₁ relative to the unprimed frame, such that dx/dt < c^2/v₁. So, for any events with a space-like separation in one frame (i.e. |dx/dt| > c), you can always find some other frame where the order of the events is reversed; on the other hand, for any events with a time-like separation (|dx/dt| < c) or a light-like separation (|dx/dt| = c), all inertial frames agree on their order.

 

[JustinLevy]

Times we give to an event are just part of a coordinate system choice.

We can always relabel as such:
x' = x
y' = y
z' = z
t' = -t

If the original coordinate system was an inertial coordinate system, then so to is the primed one.

Some rhetorical questions to promote thinking: Does this mean we switched cause and effect? Does this mean cause and effect are relative? My feeling is no, but I can't define cause-effect well enough to back that up.

We have an innate sense of what time "is", but this doesn't fit well with the mathematical notion of a coordinate system. Heck, I've had interesting lively discussions with other physicists on just the notion of the "arrow of time". Often it devolves to the realization we can't define cause and effect precisely enough to have a deep discussion. I think this is why so many people like the thermo arrow of time solution, since it skips that question entirely.

 

[Ken Natton]

It seems to me that a good deal of the interactions between experts and non-experts on these forums is a question of walking a fairly fine line between credulous acceptance of anything you are told and valid scepticism and challenge that enhances understanding on the one hand, and between open mindedness to new and difficult concepts and closed minded, stubborn adherence to misconceptions on the other. I accept that it is entirely we, the non-experts who must walk that line.

So it is with some trepidation that I offer a challenge when several more expert contributors offer a similar message. It is one matter to look at a particular, unfamiliar, counter-intuitive concept and recognise that one of the key obstructions to understanding it lies in the cause and effect view of things that is part of our culture and ingrained into our way of thinking from an early age. It is quite another to offer a philosophical argument that questions the validity of cause and effect ideas all together. I try to be open minded and I am willing to struggle with difficult concepts and try to gain some insight into them without expecting it to come too easily. But perhaps the most diplomatic way of phrasing it is to say that suggestions that cause and effect ideas are completely open to question causes my instinctive defences to rise. Without being too melodramatic, I might offer an example of my objections by suggesting that such a notion could have serious implications for criminal justice.

Now, clearly such an argument is entirely a philosophical one and not a scientific one. But it is perhaps the reason why I am predisposed to prefer an explanation that holds that cause and effect ideas are entirely compatible with relativity theory.

 

[JesseM]

So it is with some trepidation that I offer a challenge when several more expert contributors offer a similar message. It is one matter to look at a particular, unfamiliar, counter-intuitive concept and recognise that one of the key obstructions to understanding it lies in the cause and effect view of things that is part of our culture and ingrained into our way of thinking from an early age.

"Cause and effect" is still valid in relativity provided you assume that no causal influence can travel faster than the speed of light. Different inertial frames never disagree on the order of events with a time-like separation (meaning dx < c*dt between the events, i.e. the distance in light-years between them is smaller than the time in years between them, i.e. a signal moving slower than light could get from one event to the other), and they also never disagree on the order of events with a light-like separation (meaning dx = c*dt between the events, i.e. the distance in light-years between them is exactly equal to the time in years between them, i.e. a signal moving at exactly the speed of light could get from one event to the other). The only case where different inertial frames can disagree on the order of events is when there is a space-like separation between them (meaning dx > c*dt between the events, i.e. the distance in light-years between them is greater than the time in years between them, i.e. a signal would have to move faster than light to get from one to the other), so as long as causal influences never travel faster than light, these disagreements about order only occurs for pairs of events that can have had no causal influence on one another.

 

[bcrowell]

"Cause and effect" is still valid in relativity provided you assume that no causal influence can travel faster than the speed of light.

That's only in SR, not GR. GR allows CTCs. It's just that we don't know if any process in our universe can create CTCs.

 

[JustinLevy]

But perhaps the most diplomatic way of phrasing it is to say that suggestions that cause and effect ideas are completely open to question causes my instinctive defences to rise.

And fairly so.
In my post I didn't mean cause and effect ideas are completely open to question. After all, it is common to essentially define the "causal structure" of spacetime where two events in each other's light cones are causally related, and otherwise not. This is completely coordinate system independent. JesseM and others gave some explicit discussion on this in flat spacetime and inertial coordinate systems related by Lorentz transformations (or translations and rotations as well).

I was merely trying to point out that "ordering" requires a sense of direction of time. That is difficult to define in a coordinate independent method. There is no generic geometric way to do this just from the metric. One would have to refer to specific physics. In our specific case, maybe cosmological time given that our universe appears it will expand forever. Or maybe something from electro-weak theory as it does not have T symmetry, one could ad-hoc define something as "forward" in time. The most popular "arrow of time" solution is of course the thermodynamics one, the direction of entropy increase.

The overall point was just to help some interesting thought along the realization that our intuitive feelings of "ordering" in time can't be made rigorous just from the geometry of spacetime (which most answers here are referring to). We need to appeal to something else, along with the causal structure of spacetime.

Does that clear it up a bit?

-----

JesseM,
Assuming Ken Natton was responding in part to my previous post, I think you missed the point in your response to him.

In particular this statement:

The only case where different inertial frames can disagree on the order of events is when there is a space-like separation between them

We can always relabel as such:
x' = x
y' = y
z' = z
t' = -t

If the original coordinate system was an inertial coordinate system, then so to is the primed one. And Voila, the ordering is changed.

So while the causal structure of spacetime is important, additional ingredients must be included to obtain an "ordering".

 

[JesseM]

JesseM,
Assuming Ken Natton was responding in part to my previous post, I think you missed the point in your response to him.

In particular this statement:

We can always relabel as such:
x' = x
y' = y
z' = z
t' = -t

If the original coordinate system was an inertial coordinate system, then so to is the primed one.

Are you sure that's allowed? In the simplest form of the Lorentz transformation (where we assume both frames have their spatial axes aligned and that the primed frame moves along the x-axis of the unprimed frame), this would not be a valid case of the transformation. Do the more general Lorentz transformation equations (where the spatial axes may not be aligned and the direction of relative motion may not be along the x-axis) include this as a special case? If not then the above does not technically qualify as an "inertial frame" in SR. And if so, can you point to some source which gives a form of Lorentz transformation equations which would include the above as a special case?

Just based on the fact that there are violations of T-symmetry in the Standard model (though not of CPT-symmetry), which I'm pretty sure is considered to be a Lorentz-symmetric theory, I would suspect such a transformation is not allowed under the Lorentz transformation...

 

[JustinLevy]

Can we agree on the following?
- If you define ordering of events as the ordering of their time labels, then yes ordering is coordinate system dependent (even inside light cones).I assume we can agree on that, as I gave an explicit example above. I want to make it clear people tend to use this coordinate system dependent information to define ordering. Which, if you want ordering to be coordinate system independent you obviously then cannot do.

Once we agree on this, then the thrust of your question has much less meaning. For it just reduces to discussion over whether that specific coordinate system is an inertial coordinate system. This is a separate issue. I don't like to get into terminology disputes, so let's just define an inertial coordinate system so we each know what we mean by the term. That should clarify discussion.

When global inertial coordinate systems are possible, I would call any coordinate system that has a (-1,1,1,1) diagonal metric everywhere, (or opposite signature depending on your sign choice), an inertial coordinate system. This is essentially how Landau defines it with his homogeneous and isotropic requirement. This is also the definition wiki seems to have settled on http://en.wikipedia.org/wiki/Inertial_frame_of_reference

As an aside, you seem to be approaching this from the other side: that SR defines what counts as an inertial frame. If you approach SR like this, then it is trivially true as a tautology. The modern definition of SR as requiring the laws of physics to have Poincare symmetry, avoids the definition of inertial frame entirely and therefore, while in addition to being more mathematically rigorous, doesn't depend on a notoriously difficult concept to be defined precisely.

----
EDIT: Just read what you appended to the post. Yes the Standard model has Lorentz symmetry. And yes it doesn't have T symmetry. If it did, would you count that as an inertial frame? See why that is not how you should define an inertial frame? If you use "what spatial/time symmetries physics has" to define an inertial frame, then trivially and tautologically all inertial frames will have the physics look the same. The modern definition of SR is the laws of physics have Poincare symmetry (Lorentz symmetry + translations + rotations). So the standard model can fit with SR, and yet still look different in inertial coordinate systems with different handedness (as even hinted at by the wiki article on inertial frames).

 

[JesseM]

Can we agree on the following?
- If you define ordering of events as the ordering of their time labels, then yes ordering is coordinate system dependent (even inside light cones).

Yes, that's true, if we allow non-inertial coordinate systems.

Once we agree on this, then the thrust of your question has much less meaning. For it just reduces to discussion over whether that specific coordinate system is an inertial coordinate system. This is a separate issue. I don't like to get into terminology disputes, so let's just define an inertial frame so we each know what we mean by the term. That should clarify discussion.

When global inertial coordinate systems are possible, I would call any coordinate system that has a (-1,1,1,1) diagonal metric everywhere, (or opposite signature depending on your sign choice), an inertial coordinate system. This is essentially how Landau defines it with his homogeneous and isotropic requirement. This is also the definition wiki seems to have settled on http://en.wikipedia.org/wiki/Inertial_frame_of_reference

Wikipedia isn't necessarily a trustworthy source, and I'd want to know more about why you add the qualifier "essentially" to your comment about Landau. I think any really rigorous definition of inertial frames would not say that the Standard Model violates the postulate of SR that says the laws of physics are the same in all inertial frames, despite the fact that the Standard Model is not T-symmetric (see below)

EDIT: Just read what you appended to the post. Yes the Standard model has Lorentz symmetry. I don't understand how you are approaching defining an inertial frame here.

Do you agree the Standard Model is believed to violate T-symmetry (see the second-to-last paragraph http://www.lbl.gov/abc/wallchart/chapters/05/2.html), and that this means that if you write down its equations in one inertial frame with coordinates x,y,z,t, and then transformed the equations according to the transformation:

x'=x
y'=y
z'=z
t'=-t

...then you would get a slightly different set of equations, i.e. the laws of physics would not be invariant under this transformation?

EDIT: I see you edited your own last comment to read:

If it did, would you count that as an inertial frame?

I think I would, yes.

See why that is not how you should define an inertial frame? If you use "what spatial/time symmetries physics has" to define an inertial frame, then trivially and tautologically all inertial frames will have the physics look the same.

I don't agree it's trivial. We could imagine a set of physical laws where it would not be possible to find a set of coordinate systems moving at constant coordinate velocity relative to one another which satisfied both postulates of SR.

 

[JustinLevy]

Oops, I must have editted while you posted.

You still have not defined what you mean by an inertial frame. I gave my definition, and mentioned Landau and wiki just to show it wasn't obscure. I'm not trying to argue by authority (seriously, the term "inertial frame" is quite poorly defined historically ... disagreeing on subtleties is no big deal. Once we know what each other means, we're all set.).

The "essentially" in my statement regarding Landau is that he doesn't refer to the metric, he just refers to homogenous and isotropic. The metric statement was just to make this more concrete.

Can we agree on this:
- The modern definition of SR is the laws of physics have Poincare symmetry (Lorentz symmetry + translations + rotations).

This is how many books using SR approach it. While on the other hand books trying to follow the historical tract sometimes either avoid the issue of defining an inertial frame completely or have a footnote pointing out the difficulty of doing so.

Yes the Standard model has Lorentz symmetry. And yes it doesn't have T symmetry. If it did, would you count that as an inertial frame? See why that is not how you should define an inertial frame? If you use "what spatial/time symmetries physics has" to define an inertial frame, then trivially and tautologically all inertial frames will have the physics look the same. The modern definition of SR is the laws of physics have Poincare symmetry (Lorentz symmetry + translations + rotations). So the standard model can fit with SR, and yet still look different in inertial coordinate systems with different handedness (as even hinted at by the wiki article on inertial frames).

The question of: Does the standard model agree with SR? means to a physicist, Does the standard model have Poincare symmetry?

--

This is getting off topic. We already agreed time ordering is coordinate dependent even for events in a light cone. This is enough to demonstrate the main point:

So while the causal structure of spacetime is important, additional ingredients must be included to obtain an "ordering".

 

[JesseM]

This is getting off topic. We already agreed time ordering is coordinate dependent even for events in a light cone. This is enough to demonstrate the main point:

So while the causal structure of spacetime is important, additional ingredients must be included to obtain an "ordering".

I agree, we're not debating any real physical question, just which definition is "best". My opinion can be summed up by the idea that it's "best" if we choose a definition of inertial frames such that the two basic postulates of SR can still be satisfied by non-T-symmetric theories like the Standard Model. It's not a very important point, though.

 

[Ken Natton]

Unfortunately, I have limited time to post right now, but I did want to respond quickly to the conversation between JesseM and JustinLevy. In my previous post, I was addressing everyone who had posted on this thread ahead of me, because the message I was taking from all of you was that question marks over cause and effect had some validity. I think your subsequent conversation, apart from being of great interest to me, has addressed the concern I expressed, and into the bargain you have successfully brought the thread back from the philosophical edge I had pushed it towards, back into a more scientific vein.

My struggle is with the distinction you have identified between special relativity and general relativity. Of course I understand that the mathematical formalism of general relativity is much more complex than that of special relativity, and I confess that its concepts stretch me. But my understanding was that, to a significant degree, general relativity is just an extension of the basic ideas of special relativity to cover all reference frames. So it doesn’t entirely make sense to me that time ordering can be affected by a non-inertial reference frame where it cannot by an inertial reference frame. But I am also aware that I am in greater danger of being guilty of the close mindedness I spoke of before. This is now a conversation that I cannot add much to, but I expect to gain quite a bit from following it.

 

[jcsd]

Unfortunately, I have limited time to post right now, but I did want to respond quickly to the conversation between JesseM and JustinLevy. In my previous post, I was addressing everyone who had posted on this thread ahead of me, because the message I was taking from all of you was that question marks over cause and effect had some validity. I think your subsequent conversation, apart from being of great interest to me, has addressed the concern I expressed, and into the bargain you have successfully brought the thread back from the philosophical edge I had pushed it towards, back into a more scientific vein.

My struggle is with the distinction you have identified between special relativity and general relativity. Of course I understand that the mathematical formalism of general relativity is much more complex than that of special relativity, and I confess that its concepts stretch me. But my understanding was that, to a significant degree, general relativity is just an extension of the basic ideas of special relativity to cover all reference frames. So it doesn’t entirely make sense to me that time ordering can be affected by a non-inertial reference frame where it cannot by an inertial reference frame. But I am also aware that I am in greater danger of being guilty of the close mindedness I spoke of before. This is now a conversation that I cannot add much to, but I expect to gain quite a bit from following it.

Special relativity mathematically takes place against a background of Minkowski space. General relativity takes place against a background of 4 dimensional Lorentzian manifolds of which Minkwoski space is one (special) example. The motivation for this is that choosing Lorentzian manifolds other than Minkowski space allows gravity to be modeled relativistically.

Minkowski space naturally lends itself to a certain class of coordinate system called Minkowski coordinates. Physically these coordiante systems can be thoguht of as representing inertial frames of reference. However in general Lorentzian manifolds do not lend themselves naturally to (global) coordiante systems, so whilst special relativity can be explained pretty well in global Minkwoski coordinates, there's no such equivalent set of global coordinate systems to explain general relativity in, so a more general explanation that does not limit itself to certain coordinate systems is required.

So general relativity is so much more than generalizing special relativity to all coordinate systems. Many would say that modelling non-inertial observers in Minkowski spacetime is still part of special relativity.

Time ordering events is entirely dependent on how you choose to paramterize time and parametrizing time in spacetime is the job of a coordinate system in spacetime. So it should be no surprise that different choices of coordinate systems can lead to different time orderings for events

 

[bcrowell]

My struggle is with the distinction you have identified between special relativity and general relativity. Of course I understand that the mathematical formalism of general relativity is much more complex than that of special relativity, and I confess that its concepts stretch me. But my understanding was that, to a significant degree, general relativity is just an extension of the basic ideas of special relativity to cover all reference frames. So it doesn’t entirely make sense to me that time ordering can be affected by a non-inertial reference frame where it cannot by an inertial reference frame.

The way I would put it is that Einstein's goal with GR was to make a relativistic theory of gravity, and one of the insights he had that allowed him to get to that goal was realizing that there were logical problems with defining the difference between a nonaccelerating frame and an accelerating one, so that the distinction should be abandoned. But that's not the main event. The main event is depicting gravity in terms of curved spacetime.

In SR, spacetime is like a flat piece of paper, covered with a graph-paper grid having axes for time t and position x. (Really there are three spatial dimensions, but let's ignore y and z for now.) In GR, the piece of paper can be curved. If you can curve the paper, then clearly you can have cases where the time axis wraps around on itself in a circle, in the same way that lines of latitude and longitude on the Earth's surface do. That's what a CTC is. Whether our universe ever actually does this is a whole different issue.

 

[Rasalhague]

In SR, spacetime is like a flat piece of paper, covered with a graph-paper grid having axes for time t and position x. (Really there are three spatial dimensions, but let's ignore y and z for now.) In GR, the piece of paper can be curved. If you can curve the paper, then clearly you can have cases where the time axis wraps around on itself in a circle, in the same way that lines of latitude and longitude on the Earth's surface do. That's what a CTC is. Whether our universe ever actually does this is a whole different issue.

A cylinder is curved in one sense, but I gather in this context it would be said to have no intrinsic(?) curvature. Would it be hypothetically possible to have something analogous to a cylinder in spacetime, a geometry that wraps around on itself even though it's flat, and could CTCs exist in such a shape? Does gravity have anything to do with curvature in that everyday sense of "it can wrap around on itself" as well as with the kind of curvature that distinguishes a sphere, on the one hand, from a flat piece of paper and a cylinder, on the other?

And is there a name for the cylindrical "wrap-around" kind of curvature which doesn't depend on an embedding, and so is intrinsic, but not in the same way as the kind of curvature that prevents us from using global Cartesian coordinates?

 

[bcrowell]

A cylinder is curved in one sense, but I gather in this context it would be said to have no intrinsic(?) curvature. Would it be hypothetically possible to have something analogous to a cylinder in spacetime, a geometry that wraps around on itself even though it's flat, and could CTCs exist in such a shape?

Yes. If you simply take Minkowski space and identify the surface $t=t_1$ with the surface $t=t_2$, then you have a spacetime that has CTCs and zero intrinsic curvature everywhere.

Does gravity have anything to do with curvature in that everyday sense of "it can wrap around on itself" as well as with the kind of curvature that distinguishes a sphere, on the one hand, from a flat piece of paper and a cylinder, on the other?

The Einstein field equations describe gravity as a relationship between mass-energy (actually the stress-energy tensor) and a certain type of intrinsic curvature (Ricci curvature, basically the part of the spacetime curvature that isn't a tidal curvature due to distant masses). You can have a wrap-around topology in space or time without having any gravity going on at all, but having gravity increases the number of ways you can do it. The field equations are local, so they have nothing to say directly about global features like topology. However, there are mathematical links between local stuff (curvature) and global stuff (like topology).

And is there a name for the cylindrical "wrap-around" kind of curvature which doesn't depend on an embedding, and so is intrinsic, but not in the same way as the kind of curvature that prevents us from using global Cartesian coordinates?

I wouldn't refer to the cylindrical case using the word "curvature" at all. It has no intrinsic curvature.

 

[JesseM]

And is there a name for the cylindrical "wrap-around" kind of curvature which doesn't depend on an embedding, and so is intrinsic, but not in the same way as the kind of curvature that prevents us from using global Cartesian coordinates?

I wouldn't refer to the cylindrical case using the word "curvature" at all. It has no intrinsic curvature.

Yeah, here the word Rasalhague is looking for is probably "topology". Here is a good short article on the possibility that space could "wrap around" if it had a nontrivial topology, and time could theoretically do something similar in SR.

 

[Al68]

In SR, for any two events A and B that are time like separated, if A is earlier than B in one frame, it is earlier than B in all frames.

This is correct for inertial reference frames, but not true in general for arbitrary reference frames. Consider a ship moving toward Earth at 0.8c that decelerates to come to rest with Earth at a distance of 10 light years. Using the standard SR simultaneity convention, wrt the ship observer, people who were dead on Earth simultaneous with the ship beginning deceleration are alive on Earth simultaneous with the ship stopping its deceleration.

Yep, dead people can rise from the grave in SR.

 

[yossell]

This is correct for inertial reference frames, but not true in general for arbitrary reference frames. Consider a ship moving toward Earth at 0.8c that decelerates to come to rest with Earth at a distance of 10 light years. Using the standard SR simultaneity convention, wrt the ship observer, people who were dead on Earth simultaneous with the ship beginning deceleration are alive on Earth simultaneous with the ship stopping its deceleration.

Yep, dead people can rise from the grave in SR.

Ok, thanks, good point - I'll be more careful to write `inertial frame' instead of just `frame' in future.

Having said that, and I ask out of interest rather than feistiness, and because this would be a very cool example, what's the more general definition of a frame? Accelerating observers who construct frames using that simultaneity convention can't define 1-1 continuous mappings from R^4 to all of Minkowski space-time; does a frame drop the idea that it's a global mapping? (that's what happens in GR). I'm trying to think of a coordinate patch which connects the rocket at the beginning of deceleration and the Earth when the people are dead - for temporal comparison - and also connects the rocket at the end of deceleration with the Earth when the people are alive. I can't quite yet convince myself that's it's possible, but I don't want to spend too much time if you had another idea in mind.

 

[starthaus]

This is correct for inertial reference frames, but not true in general for arbitrary reference frames. Consider a ship moving toward Earth at 0.8c that decelerates to come to rest with Earth at a distance of 10 light years. Using the standard SR simultaneity convention, wrt the ship observer, people who were dead on Earth simultaneous with the ship beginning deceleration are alive on Earth simultaneous with the ship stopping its deceleration.

Yep, dead people can rise from the grave in SR.

Do you have a reference for that? My calculations show that your claim is false. The line of simultaneity keeps rotating wrt the x-axis in the Minkowski diagram as the speed varies but it never crosses over itself, even at the inflection points.

 

[Rasalhague]

Do you have a reference for that? My calculations show that your claim is false. The line of simultaneity keeps rotating wrt the x-axis in the Minkowski diagram as the speed varies but it never crosses over itself, even at the inflection points.

I get 10*Sinh[ArcTanh[.8]] = 10*0.8/Sqrt[1 - 0.8^2] = 13 + 1/3 years. So, unless I'm mistaken, people dead for less than this time, in the rest frame of the ship before it decelerates, will be alive still in the rest frame of the ship after it decelerates, i.e. in the rest frame of the earth.

EDIT: Correction: that should be 10*Sinh[ArcTanh[.8]]/Cosh[ArcTanh[.8]] = 10*Tanh[ArcTanh[0.8]] = 10*Sqrt[1 - 0.8^2]*0.8/Sqrt[1 - 0.8^2] = 10*0.8 = 8 years. I forgot to first convert the distance of 10 light years to 6, as it is in the rest frame of the ship before it decelerates. (Thanks for pointing that out, Al68.)

 

[yossell]

Do you have a reference for that? My calculations show that your claim is false. The line of simultaneity keeps rotating wrt the x-axis in the Minkowski diagram as the speed varies but it never crosses over itself, even at the inflection points.

What are your calculations? I have a counter-argument:

Restricting all de- and acceleration to the x-axis, to the 2-d case, consider a curving world-line w, and consider any two lines of simultaneity l1 and l2 of w that intersect w at p1 and p2. If these lines are not parallel, then l1 and l2 must at some point intersect and cross, say at point z. Thus lines of simultaneity of an accelerating object must intersect and cross.

 

[starthaus]

What are your calculations? I have a counter-argument:

Restricting all de- and acceleration to the x-axis, to the 2-d case, consider a curving world-line w, and consider any two lines of simultaneity l1 and l2 of w that intersect w at p1 and p2. If these lines are not parallel, then l1 and l2 must at some point intersect and cross, say at point z.

What if they intersect to the right of the wordline? (i..e. their extensions intersect)

Thus lines of simultaneity of an accelerating object must intersect and cross.

I want to see Al68's reference.