Is event sequencing relative?

[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 realise 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 (http://en.wikipedia.org/wiki/T-symmetry) (see the second-to-last paragraph here (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 modelled 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 suprise 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 (http://plus.maths.org/issue10/features/topology/) 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. :cool:

[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. :cool:

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. :cool:

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.

[yossell]

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


I don't know what you're asking for here. But by changing the direction of acceleration, you can get an intersection on either `side' of the world line.

Choose a frame, let rocket and person begin stationary in this frame, with the human remaining and having been stationary in this frame. Human having just died at (0, 0, 0, 0), while the rocket is at (0, X, 0, 0) (coordinates of the form (t, x, y, z).

Let rocket quickly accelerate away in the positive x direction, reaching speed v at time t (in stationary frame) in positive x direction and then maintaining v. Its lines of simultaneity are now tilted, and simply by letting X and v be big enough and t small enough, arbitrary events of coordinate (-T, 0, 0, 0) intersect its simultaneity line. For some of these events, the human is not a corpse.

[starthaus]

I don't know what you're asking for here. But by changing the direction of acceleration, you can get an intersection on either `side' of the world line.

Choose a frame, let rocket and person begin stationary in this frame, with the human remaining and having been stationary in this frame. Human having just died at (0, 0, 0, 0), while the rocket is at (0, X, 0, 0) (coordinates of the form (t, x, y, z).

Let rocket quickly accelerate away in the positive x direction, reaching speed v at time t (in stationary frame) in positive x direction and then maintaining v. Its lines of simultaneity are now tilted,

Yes, they start at 0 degrees wrt x and they tilt continously (with an asymptote angle of \pi/4 )as the rocket accelerates. Yet, starting point of each line of simultaneity also moves since the rocket is tracing the wordline.



and simply by letting X and v be big enough and t small enough, arbitrary events of coordinate (-T, 0, 0, 0) intersect its simultaneity line. For some of these events, the human is not a corpse.

Try proving this mathematically.

[yossell]

Yes, they start at 0 degrees wrt x and they tilt continously (with an asymptote angle of )as the rocket accelerates. Yet, starting point of each line of simultaneity also moves since the rocket is tracing the wordline.

But although the line of simultaneity has moved on, the lines are indefinitely extended in the x and -x directions. The equations mx + c and m'x + c' always have a solution if m and m' are distinct, no matter how big c - c' is.

Try proving this mathematically

Ok - I'll give it a shot - but what will you allow me? Can I work in natural units where lightrays equation is t = x and t = -x, and where objects with velocity v has a simultaneity line of 1/v? Or does this need to be established too?

[Al68]

Do you have a reference for that? Sure: http://www.bartleby.com/173/11.html. That's just a reference for the lorentz transformations. I assume you know how to use them.

Assuming that the ship starts a (short duration) decel simultaneous with the 2020 Presidential election in the inertial frame of the ship prior to decel, it will be 2012 on earth simultaneous with the decel in the inertial frame of the ship after decel.

Notice that earth's clock "jumps backward" in my example for the same reason that earth's clock "jumps ahead" in the standard twins paradox resolutions. It's just an artifact of using the SR simultaneity convention to assign coordinates to distant events.

[yossell]

starthaus, I know you're the hard man of forum and that you run a tight ship, but won't you take a look at this diagram? Red dotted line a null beam, green arrow path of rocket. It starts stationary and then gently accelerates to a velocity where the slanty line is its line of simultaneity. As X is general, it's easy to find an X where it can gently accelerate to a v and a point where the line of simultaneity hits (-T 0 0 0)

[bcrowell]

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. :cool:

The reason we care about frame-independence of time ordering in SR is that it's necessary for causality. Your example refers to simultaneity between events that are spacelike separated, so it doesn't have any implications for causality. Depending on how you interpret yossell's statement of the principle, I'm not even sure that your example is a counterexample.

Here is a somewhat different statement of the principle that may be more clearcut. Let A and B be the end-points of a curve S in spacetime, such that the curve's orientation is always in the positive timelike direction as we traverse it from A to B. If the spacetime is flat, then this property of S is coordinate-independent.

[starthaus]

starthaus, I know you're the hard man of forum and that you run a tight ship,

Yes, I do :-)


but won't you take a look at this diagram? Red dotted line a null beam, green arrow path of rocket. It starts stationary and then gently accelerates to a velocity where the slanty line is its line of simultaneity. As X is general, it's easy to find an X where it can gently accelerate to a v and a point where the line of simultaneity hits (-T 0 0 0)

bcrowell beat me to the disproof but I can disprove your statement mathematically (Al68 offered nothing but some handwaving). Are you interested in the mathematical disprroof?

[yossell]

bcrowell beat me to it but I can disprove your statement mathematically (Al68 offered nothing but some handwaving). Are you interested in the mathematical disprroof?

I'd be interested, but I now think we must be talking at cross-purposes. I (and I think Al68), are considering examples where an object, by accelerating or decelerating, changes inertial frames in a way which affects the T-coordinate of distant events - bcrowell is talking about the property of particular CURVES in spacetime, and the question of whether they are timelike or not is not a coordinate dependent matter. This latter I think is true, and am not arguing against it.

[Rasalhague]

In #33, I made a calculation based on my interpretation of Al68's resurrection statement. Is this what you had in mind, Al68? Are other people here interpreting it differently? Did I get the calculation right?

[Al68]

The reason we care about frame-independence of time ordering in SR is that it's necessary for causality. Your example refers to simultaneity between events that are spacelike separated, so it doesn't have any implications for causality. Depending on how you interpret yossell's statement of the principle, I'm not even sure that your example is a counterexample.You're right, it's not a counterexample (to causality), yossel clearly meant to refer strictly to inertial reference frames, not an arbitrary coordinate system. And you're right that there are no implications for causality. Assigning coordinates to distant events with the SR simultaneity convention doesn't cause anything.

[Al68]

In #33, I made a calculation based on my interpretation of Al68's resurrection statement. Is this what you had in mind, Al68? Are other people here interpreting it differently? Did I get the calculation right?Yeah your interpretation is right, but I think you missed that the coordinate distance between the ship and earth is only 6 ly in the ship's frame prior to decel. So it should be 8 yrs instead of 13.33 yrs.

[Al68]

...Al68 offered nothing but some handwaving...LOL. You didn't ask for a proof. You asked for the reference I used. I provided it.

Edit: Rasalhague shows the math in the next post, but I assumed an experimental physicist like yourself wouldn't need it to be shown.

[Rasalhague]

Yeah your interpretation is right, but I think you missed that the coordinate distance between the ship and earth is only 6 ly in the ship's frame prior to decel. So it should be 8 yrs instead of 13.33 yrs.

Oh yeah, of course! 10*Sqrt[1-0.8^2]*0.8/Sqrt[1-0.8^2] = 10*0.8 = 8. Sorry deads! At least we get some back, and with a bigger boost as many as we want...

[Al68]

Oh yeah, of course! 10*Sqrt[1-0.8^2]*0.8/Sqrt[1-0.8^2] = 10*0.8 = 8. Sorry deads! At least we get some back, and with a bigger boost as many as we want...Yep. Too bad the ship's observer can never actually observe them alive again. They're alive "now" according to the SR simultaneity convention, but no "they're alive" signal will ever reach the ship "later" regardless of its subsequent motion. In fact, the faster the ship tries to reach earth, the quicker they will die off again. :frown:

[starthaus]

LOL. You didn't ask for a proof. You asked for the reference I used. I provided it.

I asked you for a valid reference to your claim that acceleration can produce event order reversal. So, you provided no reference. Not only that, you got the wrong answer.

[JesseM]

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.
It depends on how fast the acceleration is, i.e. how long it takes the ship to go from 0.8c to rest in the Earth frame. For example, suppose the ship is traveling towards Earth at 0.8c until it is 10 light years from Earth at t=0 years in the Earth frame (so if the Earth is at x=0 light years, the ship could be at x=-10 light years), then it accelerates for a year until it is at rest at t=1 year the Earth frame. In that case, in the ship's inertial rest frame at the moment it begins to accelerate, the moment of its beginning to accelerate is simultaneous with an event that occurs on Earth at t=8 years in the Earth frame (since in the Earth's frame we have dx=10 and dt=8 between these events, meaning in the ship's frame dt'=gamma*(dt - v*dx/c^2)=0), but in the ship's inertial rest frame at the moment it stops accelerating, the moment it stops accelerating is simultaneous with an event that occurs on Earth at t=1 year in the Earth frame.

[starthaus]

It depends on how fast the acceleration is, i.e. how long it takes the ship to go from 0.8c to rest in the Earth frame. For example, suppose the ship is traveling towards Earth at 0.8c until it is 10 light years from Earth at t=0 years in the Earth frame (so if the Earth is at x=0 light years, the ship could be at x=-10 light years), then it accelerates for a year until it is at rest at t=1 year the Earth frame. In that case, in the ship's inertial rest frame at the moment it begins to accelerate, the moment of its beginning to accelerate is simultaneous with an event that occurs on Earth at t=8 years in the Earth frame (since in the Earth's frame we have dx=10 and dt=8 between these events, meaning in the ship's frame dt'=gamma*(dt - v*dx/c^2)=0), but in the ship's inertial rest frame at the moment it stops accelerating, the moment it stops accelerating is simultaneous with an event that occurs on Earth at t=1 year in the Earth frame.

I don't think that the above is correct. The issue in discussion was whether acceleration can invert the ordering of events. I will repeat the proof I gave for inertial frames and I will generalize to accelerated frames:

dt'=\gamma(\tau-vx/c^2)

dt'=\gamma (d\tau-vdx/c^2)

dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})

Since v \frac{dx}{d\tau}<c^2 it follows that dt' and d\tau always have the same sign. If you are ok with the above, I can post the generalization to accelerated frames.

[JesseM]

I don't think that the above is correct. The issue in discussion was whether acceleration can invert the ordering of events. I will repeat the proof I gave for inertial frames and I will generalize to accelerated frames:

dt'=\gamma(\tau-vx/c^2)

dt'=\gamma (d\tau-vdx/c^2)

dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})
How would you define the non-inertial frame for a non-inertial observer? There's no single "correct" way, but I think the most common type of non-inertial frame used by physicists in SR would be one that has the following properties:

1. For point on the non-inertial observer's worldline, the coordinate time is just equal to the observer's proper time at that point

2. At any point on the observer's worldline, a line of simultaneity in the non-inertial coordinate system which goes through that point would be identical to a line of simultaneity in the inertial frame where the observer has an instantaneous velocity of zero at that point

3. For two events which lie on a single line of simultaneity in the non-inertial coordinate system, the coordinate distance between them should be the same as the coordinate distance in the inertial frame which also has a line of simultaneity going through both events

4. The non-inertial observer has a coordinate position that doesn't change with coordinate time in the non-inertial system

If you define the non-inertial coordinate system in this way, then (3) implies the system's judgments about simultaneity always agree with those of the observer's instantaneous inertial rest frame, so this was the basis for my comments above (although I think it would actually be more common to just say the coordinate system "ends" at the point where lines of simultaneity would cross over one another, as with Rindler coordinates which don't extend past the Rindler horizon). I think Al68 was also assuming this sort of coordinate system when he commented about the dead rising from the grave in post #30 (http://www.physicsforums.com/showthread.php?p=2826017#post2826017). If you're assuming a different type of non-inertial coordinate system, you'll have to explain how it's defined for an observer with a non-inertial worldline that has a known parametrization x(t) relative to some inertial frame (for an observer experiencing constant proper acceleration a, their x(t) in an inertial frame where they started at rest would be x(t) = (c^2/a) (sqrt[1 + (at/c)^2] - 1) according to the relativistic rocket page (http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html)).

[starthaus]

How would you define the non-inertial frame for a non-inertial observer?

I'll show you next. First off, do you agree that inertial motion cannot reverse the order of events between frames? Yes or no?

[JesseM]

I'll show you next. First off, do you agree that inertial motion cannot reverse the order of events between frames? Yes or no?
Can you clarify what you mean by that? It is certainly true that for events with a space-like separation (|dx| > |c*dt| in any given inertial frame), you can find a pair of inertial frames which disagree on their order, are you denying that? If you don't deny it, why doesn't this qualify as an example of "inertial motion reversing the order of events between frames" as you define this phrase?

[yossell]

dt'=\gamma(\tau-vx/c^2)

dt'=\gamma (d\tau-vdx/c^2)

dt'=\gamma d\tau (1-v/c^2*\frac{dx}{d\tau})

Since v \frac{dx}{d\tau}<c^2 it follows that dt' and d\tau always have the same sign. If you are ok with the above, I can post the generalization to accelerated frames.

Your question is whether inertial motion can invert the order of events.

I do not think you have shown this claim in its full generality, but I think you have shown a weaker one, that inertial motion cannot invert the order of timelike events.

First, an aside: what's going on in step 1 to 2 - why the replacement of x and \tau with dx and d\tau? Just notational?

Secondly, I take the strategy to be this: dt' represents the difference in time between two events A and B in one inertial frame; d\tau and dx the difference in time and the difference in x-coordinate between these two events in another inertial frame. You then try to show that d\tau and dt' have the same sign, and therefore that any two frames must agree on the temporal order of the two events.

The trouble is that dx/d\tau is not necessarily a velocity of anything - it is *just* a ratio. The events may be spacelike separated. In which case, the ratio is not less than c. In such cases, your proof can be used to show that the temporal order can be inverted.

However, when attention is restricted to events which are timelike separated, then your proof seems to go through.

[starthaus]

Your question is whether inertial motion can invert the order of events.

More precisely, if order of timelike events (see the OP) is maintained in SR. We aren't talking about spacelike events.


I do not think you have shown this claim in its full generality, but I think you have shown a weaker one, that inertial motion cannot invert the order of timelike events.

...which is precisely what I have been talking about at posts 12 and 13. This answer is for both you and JesseM. In mathematical terms, as already shown twice, the proof is for \frac{dx}{d\tau}<1 (no "faster than light signalling"). I am addressing the OP, not the case of spacelike events.



First, an aside: what's going on in step 1 to 2 - why the replacement of x and \tau with dx and d\tau? Just notational?

No, this is standard differentiation.



Secondly, I take the strategy to be this: dt' represents the difference in time between two events A and B in one inertial frame; d\tau and dx the difference in time and the difference in x-coordinate between these two events in another inertial frame. You then try to show that d\tau and dt' have the same sign, and therefore that any two frames must agree on the temporal order of the two events.

yes



The trouble is that dx/d\tau is not necessarily a velocity of anything - it is *just* a ratio. The events may be spacelike separated.

No, see above.




However, when attention is restricted to events which are timelike separated, then your proof seems to go through.

It does not "seem", it does. It is just a repeat of the one at post 12. Same conditions, same outcome.

[yossell]

I still don't understand what you're calling differentiation or what you're doing from 1 - 2.

From line 2 onwards we're agreed that it shows that, for any timelike related (and I emphasise this because you've not been explicitly putting in this rider) events, all inertial observers agree about the events' temporal order ---- as I in fact wrote way back in post 5.

Great.

Now, I'm interested in the generalisation you've mentioned, to deal with Al68's point, that a rocket ship can, beginning at space time point A and in an inertial frame where events A and space time point B are simultaneous, travel to a spacetime point C and an inertial frame where C is simultaneous with an event that lies in B's past.

[starthaus]

I still don't understand what you're calling differentiation or what you're doing from 1 - 2.

Because this is a well known operation in calculus. It is important that you understand it for the next step, accelerated motion. How familiar are you with differential calculus?

[JesseM]

...which is precisely what I have been talking about at posts 12 and 13. This answer is for both you and JesseM. In mathematical terms, as already shown twice, the proof is for \frac{dx}{d\tau}<1 (no "faster than light signalling")
OK, so when you said "First off, do you agree that inertial motion cannot reverse the order of events between frames?" were you only talking about pairs of events for which |dx/dt| <= c? If so, I agree that different inertial frames won't disagree on their order.

[starthaus]

OK, so when you said "First off, do you agree that inertial motion cannot reverse the order of events between frames?" were you only talking about pairs of events for which |dx/dt| <= c? If so, I agree that different inertial frames won't disagree on their order.

Excellent, let's wait for yossell, see if he's comfortable with some heavy duty differential calculus required for the case of accelerated frames.

Do you think that the above holds for accelerated frames?

[JesseM]

Do you think that the above holds for accelerated frames?
It depends what you allow to qualify as an accelerated frame. You can write down a coordinate transformation which relates a non-inertial coordinate system to an inertial one, with the equations of the transformation implying that lines of simultaneity in the accelerated frame will actually cross when plotted in the inertial frame, but some authors would say this is not a well-behaved frame since it doesn't assign unique coordinates to each event, so they'd require that the accelerated frame only be defined up to the point where lines of simultaneity would cross rather than throughout all of spacetime (this is what's done with Rindler coordinates (http://en.wikipedia.org/wiki/Rindler_coordinates), which only cover a region of spacetime to one side of the Rindler horizon (http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html))

Also, it depends on whether you impose the rule that the space coordinate must actually be spacelike at all times (i.e. every surface of constant time is a spacelike surface, meaning path in that surface is spacelike) and the time coordinate must be timelike everywhere (every line of constant position coordinate is a timelike worldline). If you don't impose that rule, then non-inertial frames can disagree on the order of events with a timelike separation even without any lines of simultaneity crossing in any of the non-inertial frames.

[Al68]

I asked you for a valid reference to your claim that acceleration can produce event order reversal. So, you provided no reference. Not only that, you got the wrong answer.LOL. That's not the claim I made and you know it. I said that events could be in reverse order in an accelerated reference frame. Slight difference there. And I provided the only reference I used: the lorentz transformations.

And I got the right answer. All you need to do to see how obvious it is is to imagine a clock (clock C) at rest with earth local to the ship's (almost instantaneous) deceleration, that is synched with earth's clock in earth's frame. Prior to deceleration, earth's clock is ahead of clock C by 8 years in the ship's frame. After deceleration, earth's clock matches clock C in the ship's frame. Clock C is (almost) local to the ship before and after decel and has (almost) the same reading before and after decel. Simple lorentz transformations. No kawcoolus required.

Why would an experimental physicist have so much trouble understanding such a simple scenario? What do you think the right answer is?

[starthaus]

LOL. That's not the claim I made and you know it. I said that events could be in reverse order in an accelerated reference frame.

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

[JesseM]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.
What about the accelerated frame I defined in post #53 with properties 1-4? Do you disagree that lines of simultaneity can cross in such a frame, causing the frame to label timelike-separated events with a reversed order from the order in inertial frames? Or do you think my definition wasn't clear?

[Al68]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.LOL. I can only assume my claim to be perfectly correct given that an experimental physicist such as yourself has failed to show it to be erroneous in any way. :biggrin: Thanks for the confirmation! :!!)

BTW, my claim was that (in an accelerated frame) events could be assigned time coordinates in reverse order by the SR simultaneity convention, not that they actually occur or are observed in reverse order. Unless someone was so pedantic as to take my "dead rising from the grave" statement seriously. :bugeye:

[yossell]

What about the accelerated frame I defined in post #53 with properties 1-4? Do you disagree that lines of simultaneity can cross in such a frame, causing the frame to label timelike-separated events with a reversed order from the order in inertial frames? Or do you think my definition wasn't clear?

I have a question about non-inertial frames: In the kinds of frames that you're talking about, one and the same space-time point will be assigned two different coordinates - e.g. the point of intersection of the lines of simultaneity. But does the concept of a frame allow for this to happen? The time of an event will be multivalued in such a frame. Do people know if this is ok?

Let me say that I don't see anything deeply physical going on here - I recognise that non-inertial frames are a somewhat artificial concept. it's just an interesting (to me) question about the definition of a frame.

[Ken Natton]

Wow. I can’t believe the pace that this thread has gone at. Just one day away and I’ve had four pages of challenging stuff to read and try to understand. I hate to act as a weight and drag the pace down, but my hope is that if anyone does answer me, that doesn’t necessarily have to impinge on the conversation between the rest of you.

I’m still on the difference between special and general relativity.

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 modelled relativistically.


Okay, but like Cartesian co-ordinate systems and Euclidian geometry, these metrics are just intellectual constructs, tools of analysis, rather aspects of physical reality. Yes, the reality of the curvature of spacetime is part of general relativity theory, but that is always the true physical reality, a body isn’t actually transformed from being in curved spacetine to being in flat spacetime when it ceases to accelerate and starts to travel at a constant speed. An inertial reference frame is really just an idealised state to make the concept easier to understand, as is the case for Newton’s first law, when we are asked to imagine a body in a situation without gravity and without friction. The truth, is it not, is that all real bodies, from subatomic particles to galaxies, exist in permanent non-inertial reference frames?

There are those that argue that special relativity is superseded and made redundant by general relativity. The only defence against that is that special relativity is the route in to an understanding. Special relativity is more graspable for those coming to relativity for the first time, and once you have got that, it becomes easier to extend the principle to cover all reference frames.

Yes I know Einstein was specifically interested in the equivalence of gravitational freefall and acceleration under some other force, and that this led him to the idea that mass actually warps spacetime and that this is the actual explanation for gravity. But the concept of equivalence is extendable to the equivalence of all accelerating reference frames, is it not? It seems unlikely to me that special relativity describes a physical reality that only exists in idealised conditions. Surely, the physical reality is always the same. Special relativity just covers a special case of it, general relativity generalises that principle.

So I suppose, to bring it back to the original subject of the thread, what you are telling me is that within the idealised constraints of an inertial reference frame, changing the sequence of events is not possible. Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

[Rasalhague]

This is false, the events will not be in reverse order, accelerated frames preserve event ordering, just as inertial frames do. This is the same erroneous claim you made earlier.

Is the issue here: (1) "Are there pairs of non-identical events with no defined coordinate time order or with multiple coordinate time orders in one single given accelerated coordinate system?" Al68: "Yes." starthaus: "No."

At first I thought the issue was: (2) "Does Al68's scenario correctly exemplify the fact that spacelike separated events don't have a frame-independent natural time order. (First simultaneity judgement in the instantaneous comoving inertial rest frame of the ship, 10 light years from the earth, travelling towards the earth at 0.8c relative to the earth. Second simultaneity judgement in the instantaneous comoving inertial rest frame of the ship after it decelerates till it's at rest with respect to the earth, i.e. in the rest frame of the earth.)"

I thought starthaus imagined Al68 was talking about a pair of timelike separated events. I thought you were both just talking at cross purposes, particularly as starthaus's proof consisted of using the Lorentz boost formulas to show that timelike separated events can't change order under a boost.

But if the issue is actually (1), then I would have thought the answer was no because of the definition of "coordinate system", "reference frame", "chart", specifically the requirement that the coordinate functions be, as their name suggests, functions, i.e. single-valued. Events in spacetime for which a coordinate system doesn't behave this way, I'm thinking, wouldn't belong to the domain of that particular coordinate system. In this case, if we have an accelerated coordinate system something like Rindler coordinates (as JesseM says in the final paragraph of #53), events on earth just aren't covered by these coordinates. Or, if they were, then, by definition of a chart, there'd have to be one single time order specified, even though it needen't necessarily agree with the time order of another coordinate system, and--at least for some pairs of events--not every possible coordinate system will agree.

Is that anywhere near the mark?

[yossell]

Okay, but like Cartesian co-ordinate systems and Euclidian geometry, these metrics are just intellectual constructs, tools of analysis, rather aspects of physical reality.

Since the rest of this paragraph looks right to me, there's probably nothing wrong with your understanding, but I would be careful about this way of putting it. The metrics of spacetime themselves aren't normally thought of as mere intellectual constructs. The metric is quite unlike a choice of coordinate system, and in fact conveys coordinate independent information about space-time itself, such as whether space is curved, and the coordinate-independent Minkowski `length' of a space-time curve.

There have been those (I think Poincare was one) who thought that the choice of geometry itself was as much a convention as choice of coordinate system. But I wouldn't say that this was a mainstream idea today.

The truth, is it not, is that all real bodies, from subatomic particles to galaxies, exist in permanent non-inertial reference frames?

Again, this is probable pit-nicking, but accelerated bodies can be analysed from the point of view of an inertial frame. What's (I think) true is that, the existence of gravity, inertial frames as understood in SR no longer exist globally. Rather, one can apply SR at a local level - working on a small scale, not extending your t and x coordinates too far, objects still behave approximately as SR says they do.

But the concept of equivalence is extendable to the equivalence of all accelerating reference frames, is it not?


I tentatively believe it's only to all *freely* falling frames.

It seems unlikely to me that special relativity describes a physical reality that only exists in idealised conditions. Surely, the physical reality is always the same. Special relativity just covers a special case of it, general relativity generalises that principle.

Since gravity is pervasive, all space time is warped a little - so I think in that sense SR does only describe a physical reality that exists in idealised conditions. At least, it's not clear to me that there's anywhere where the idealised conditions obtain. Do you see this as a problem? The way in which SR approximates GR is mathematically well defined and well understood.

Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

I don't think this was quite the lesson - though as you can see, there was controversy and...maybe something more...

I would summarise the main points as: (a) the temporal order of two space-like events is dependent on inertial frame, but, in standard interpretations of SR*, this has no causal implications as such events lie outside the light cone and thus are causally independent; (b) very crazy/artificial non-inertial frames may be constructed on which a later event (my death) has a smaller coordinate time than an earlier event (my birth); but such frames are so artificial - really little more than a choice of labelling or giving coordinates to distant events - that nobody should try and read off the causal story or physical story off the numbers of the resulting chart; (c) some very strange solutions of GR are possible, which allow a kind of circular causation, but these solutions seem removed from reality and, at least locally, there's an event by event causal story - it's just that it loops around on itself; Even in this model, though, this circular causal chain is not a frame dependent matter.

tl;dr
I don't think you should worry.

*e.g. no tachyons.

[seto6]

if a happens first and causes b to happen later in all frame we would see a then b. the events must have their order in all frame.

[bcrowell]

if a happens first and causes b to happen later in all frame we would see a then b. the events must have their order in all frame.

It would be nice if relativity worked that way, but it doesn't. The field equations of GR admit solutions with closed timelike curves. In a spacetime with CTCs, you can't even define an ordering, much less ensure that it's coordinate-independent.

Re the discussion of accelerating frames, I'm skeptical that any of it has any significance. Changing frames of reference is just a change of coordinates. By a change in coordinates, I can always trivially reorder events in the sense of reversing the numerical ordering of their time coordinates. For example, I can simply do the coordinate transformation t \rightarrow -t. The only thing that has a coordinate-independent significance is a closed timelike curve.

[JesseM]

It would be nice if relativity worked that way, but it doesn't. The field equations of GR admit solutions with closed timelike curves. In a spacetime with CTCs, you can't even define an ordering, much less ensure that it's coordinate-independent.
This is true, but one thing to note about this is that a lot of the CTC solutions require an infinite universe that has some "unrealistic" properties throughout, like a dense rotating cylinder of infinite length (the Tipler cylinder (http://en.wikipedia.org/wiki/Tipler_cylinder)) or for the entire universe to have some nonzero rotation (the Godel metric (http://en.wikipedia.org/wiki/Gödel_metric), discussed here (http://web.archive.org/web/20070701033428/http://www.ettnet.se/~egils/essay/essay.html)). If you want to create a finite region where CTCs are allowed in an otherwise "normal" universe, like time travel based on a traversable wormhole (http://en.wikipedia.org/wiki/Wormhole#Traversable_wormholes), a result by Hawking proved that you must use exotic matter (http://en.wikipedia.org/wiki/Exotic_matter) which violates the "weak energy condition" (see third paragraph here (http://en.wikipedia.org/wiki/Time_travel#Other_approaches_based_on_general_rela tivity)), and at least in the case of wormholes some other energy conditions need to be violated too (see here (http://en.wikipedia.org/wiki/Wormhole#Raychaudhuri.27s_theorem_and_exotic_matte r), and note that quantum effects like the Casimir effect may not be sufficient). It's not known whether matter or fields that violate all these energy conditions are actually allowed by the fundamental laws of nature, so GR solutions involving them may not correspond to anything that could be realized in nature, even in principle (and this is before we get into the issue of whether CTC solutions might be one where GR's predictions would depart significantly from those of a theory of quantum gravity--some analysis (http://arxiv.org/abs/gr-qc/0204022) suggests that in semiclassical gravity (http://en.wikipedia.org/wiki/Semiclassical_gravity) the energy density of quantum fields would always go to infinity on the boundary between the CTC region and the non-CTC region, which would indicate this is a situation where semiclassical gravity breaks down and a full theory of quantum gravity is needed)

[bcrowell]

@JesseM: It seems like we have two parallel threads going on here, one on SR and one on GR, which is making it hard to keep the discussion coherent. The SR posts are swamping the GR posts, and therefore a lot of us posting on GR are repeating ourselves or repeating each other. I'm going to start a separate thread for the GR stuff, and I hope you don't mind if I quote your (very interesting!) post #73 there in full.

[JesseM]

@JesseM: It seems like we have two parallel threads going on here, one on SR and one on GR, which is making it hard to keep the discussion coherent. The SR posts are swamping the GR posts, and therefore a lot of us posting on GR are repeating ourselves or repeating each other. I'm going to start a separate thread for the GR stuff, and I hope you don't mind if I quote your (very interesting!) post #73 there in full.
Sure, go for it.

[yuiop]

So I suppose, to bring it back to the original subject of the thread, what you are telling me is that within the idealised constraints of an inertial reference frame, changing the sequence of events is not possible. Physical reality does not actually impose those constraints, and thus relativity of sequence is always possible. Once again, we have the undermining of the notion of cause and effect that worried me.

In the strictly flat space SR context, it is possible for two inertial observers to have a different opinion about the order of two events, if the two events do not share the same light cone. In this special case it is not possible for one of these events to be the cause of the other, so this case has no sigificance on the notion of cause and effect. If the events are both located in a common light cone and so one event could in principle be the cause of the other, then all inertial observers will agree on which event came first.

Things are a bit different in GR. Things are lot more complicated and I am sure you will find some experts say CTCs, time travel, hyperspace jumping to distant galaxies (or even other universes) via black holes and worm holes is possible and another group of experts that would disagree.

[yuiop]

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.

Can someone explain how the above works? How is it possible to have CTCs (i.e. return to a time in the past) in flat space?

Is just a case of relabelling time coordinates so that you effectively call tomorrow, yesterday, but no actual time travel or reversal of causality has really occured?

[yossell]

Can someone explain how the above works? How is it possible to have CTCs (i.e. return to a time in the past) in flat space?


My guess is that it's like the construction of a cylinder from a flat piece of paper by identifying the lines y = 0 and y = 1, except that its two lines of simultaneity that are identified. This is only a change of topology - intrinsically, the cylinder is still a flat surface, the Euclidean distances are still pythagorean. So the construction is compatible with the metric being flat.

I've not seen it before though - very interesting - so I'm not immediately sure whether it works in Minkowski space-time, or whether there's some hidden problem.

[yossell]

Can someone explain how the above works? How is it possible to have CTCs (i.e. return to a time in the past) in flat space?


The reason this is not obviously coherent to me (bracketing the CTCs) is this: the surfaces picked out are dependent on a frame. In different frames, different lines of simultaneity. The simultaneity lines that are not parallel to this surface will, it seems, repeatedly curl around this surface, as the x-coordinate is not bounded, and the t-lines will loop around oddly too.

Is this just odd, or is it somehow in conflict with SR?

[JesseM]

The reason this is not obviously coherent to me (bracketing the CTCs) is this: the surfaces picked out are dependent on a frame. In different frames, different lines of simultaneity. The simultaneity lines that are not parallel to this surface will, it seems, repeatedly curl around this surface, as the x-coordinate is not bounded, and the t-lines will loop around oddly too.

Is this just odd, or is it somehow in conflict with SR?
A universe with a closed spatial topology (so if you travel far enough in any direction you return to your place of origin) can have what seems to be a preferred global frame even if in any small region of spacetime the laws of physics work the same in any frame (see this thread (http://www.physicsforums.com/showthread.php?t=375432)), so I think the same would apply here. I don't really see this as a conflict with SR but I guess it depends on how you define "SR".

[Al68]

I have a question about non-inertial frames: In the kinds of frames that you're talking about, one and the same space-time point will be assigned two different coordinates - e.g. the point of intersection of the lines of simultaneity. But does the concept of a frame allow for this to happen? The time of an event will be multivalued in such a frame. Do people know if this is ok?A single event could have several time coordinates in an accelerated frame. If the dead can rise from the grave once, why not several times? :biggrin:

[Rasalhague]

A single event could have several time coordinates in an accelerated frame. If the dead can rise from the grave once, why not several times? :biggrin:

The definitions I've read of a manifold include the idea of "coordinate functions" which each associate each point in their domain with (in the case of a real manifold) a single real number. A function (map, mapping) is usually defined to exclude multi-valuedness. So if I've understood this right, a single event could have no more than one time coordinate in a single, given frame (chart, coordinate system), although there might be any number of charts that include that event in their domain, and they won't in general agree on its time coordinate.

http://mathworld.wolfram.com/CoordinateChart.html

I think the other side of the Rindler horizon is like the north pole in this example, not part of U, the domain of the chart.