Can different paths in spacetime have the same separation?

[JesseM]

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.

By "point Z" do you mean a point in space (which persists over time) or a point in spacetime (an instantaneously brief localized event)? When physicists talk about a "spacelike separation" between points, they're talking about points in spacetime. If this is what you're doing, what two events are you saying had a spacelike separation? Also, note that although in SR one can talk about the separation between two points (with it understood that we're looking at a straight line in spacetime between them), in GR one really needs to specify a path through spacetime to say if it's spacelike or timelike or lightlike, since there are multiple paths between any given pair of events and none of them uniquely represent "the" separation between them.

This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initialy at X ] is on the way to Y!

The observer ,when he arrives at Y is amazed to find that the path ahead of him has become timelike!

What do you mean "the path ahead of him"? "Ahead" in what sense? Ahead in some spatial direction, or ahead in his future light cone? Your scenario is really difficult to understand in words and I doubt anyone else is understanding it much better than I am, it would help if you either drew a spacetime diagram or gave a numerical example, or at least described all the worldlines and events more carefully, being sure to distinguish between points and paths in space and points and paths in spacetime, and perhaps also specifying which points in spacetime like in the future light cones of other points and which pairs of points are not in each other's past or future light cones (so there is no timelike path through spacetime between them, and no signal traveling at the speed of light or slower could get from one point to the other).

Observers standing at the same spatial point have the same notion at time=t2
[Spacetime point Y is the same for all observers instantaneously,when the moving observer arrives there]

You said before y was a spatial position, now you're saying it's a spacetime point? They are two very different concepts? Again, with a spatial position you can talk about the same position at different times, but a spacetime point is an instantaneously brief event.

 

[Anamitra]

I believe that I have been misread once more. So I am trying to clarify my stand:

WE consider a spacetime curve running from X[t1,x1,y1,z1] to Z[t3,x3,y3,z3] via Y[t2,x2,y2,z2]. The curve between X and Y is time like and the curve between Y and Z is EXPECTEDLY spacelike . t1<t2

In the time t1 to t2 the observer reaches from X to Y along the timelike curve with the expectation that he will find a space like curve between Y and Z.[or may be observers at the spatial position of Y standing for a long time before his advent will inform him about the nature of the path ahead of him--and how it has changed]. The apparently unreachable spacetime point is now reachable!

The term EXPECTEDLY has been explained below:

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.They could well expect the path remain spacelike for t=t2. That is the observes are expecting a spacelike separation between Y and Z which are spacetime points.This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initially at X ] is on the way to Y!

The observer ,when he arrives at Y is amazed to find that the path ahead of him has become timelike!
Observers standing at the same spatial point have the same notion at time=t2
[Spacetime point Y is the same for all observers instantaneously,when the moving observer arrives there]

 

[Anamitra]

One may,in many circumstances envisage[they may pre-calculate or they may be pre-informed] the nature of separation between a pair of spacetime points[events] before the occurrence of the events, from the present situation of the metric coefficients--their expressions/values.But when the events occur they may find that the nature of separation has changed---spacelike paths have time like or vice-versa.

 

[JesseM]

The term EXPECTEDLY has been explained below:

You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.

A spacelike separation from what?

They could well expect the path remain spacelike for t=t2.

What is this "path" you are talking about? What set of events does it pass through? You say "remain spacelike", does that mean some section of this path is spacelike, and if so what section is that? You never define any of your terms clearly!

That is the observes are expecting a spacelike separation between Y and Z which are spacetime points.

Why do they "expect" that? Are they ignorant of the metric?

This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initially at X ] is on the way to Y!

What information?

 

[PAllen]

One may,in many circumstances envisage the nature of separation between a pair of spacetime points[events] before the occurrence of the events, from the present situation of the metric coefficients--their expressions/values.But when the events occur they may find that the nature of separation has changed---spacelike paths have time like or vice-versa.

This really makes no sense. You are looking at time varying metric components expressed in some specific coordinates, and trying to explain it this way. This is not a correct explanation. A correct explanation of how this situation would be perceived is given in my post #105.

 

[Anamitra]

Please go through the posts #107 and #108, Jesse

 

[JesseM]

One may,in many circumstances envisage the nature of separation between a pair of spacetime points[events] before the occurrence of the events, from the present situation of the metric coefficients--their expressions/values.But when the events occur they may find that the nature of separation has changed---spacelike paths have time like or vice-versa.

In GR you can predict what the metric coefficients will be in the future if you know the coefficients along with the distribution of matter/energy in the present. And in any case I don't think you can meaningfully define a coordinate system on a region of spacetime where you don't know the metric, and without a coordinate system how can you even pinpoint specific events and paths in this region in order to talk about whether the paths to the events are spacelike or timelike?

 

[JesseM]

Please go through the posts #107 and #108, Jesse

My post #109 was directly in response to #107, I asked questions because I found your description there completely unclear (and I doubt anyone else reading this thread could follow it either). Post #108 is unclear as well, see my response in #112. If you want to be understood, it might help if you would try your best to give specific answers to the questions I ask.

 

[Anamitra]

In GR you can predict what the metric coefficients will be in the future if you know the coefficients along with the distribution of matter/energy in the present. And in any case I don't think you can meaningfully define a coordinate system on a region of spacetime where you don't know the metric, and without a coordinate system how can you even pinpoint specific events and paths in this region in order to talk about whether the paths to the events are spacelike or timelike?

The important aspect to consider is the finite speed of signal transmission!
I can always get informed about the nature of metrics at different points.Information about the changed state /changed values of the metrics will reach me later. There is a period of ignorance which may be a hundred years or more

[At each spatial point we may preserve information artificially or naturally for a future assessment----one may assume that]

 

[JesseM]

The important aspect to consider is the finite speed of signal transmission!
I can always get informed about the nature of metrics at different points.Information about the changed state /changed values of the metrics will reach me later. There is a period of ignorance which may be a hundred years or more

Again, you can predict what the metric will be like even in regions you can't get an actual signal from. If you don't know enough about a region to predict the metric, then it seems to me the only "events" you can talk about in this region are fairly generic ones like "the future decay of some particle I saw earlier" or "the event of some clock showing a time T later than the time it showed the most recent moment I saw it". How would you even describe a specific path to speculate about whether it'll be spacelike or timelike in that region?

I suppose if you had a family of clocks connected by flexible springs filling space, then you could define a coordinate system where each clock had a constant position coordinate and its reading defined a time coordinate, so then you could talk about coordinates even in regions where you didn't know the metric, and define a "path" in terms of those coordinates...but if you didn't know the metric it's hard to see how you could have much basis for "expecting" that a particular path would be spacelike or timelike (except a special cases like a path of constant position coordinate, which would just be the worldline of some clock and must therefore be timelike)

 

[Anamitra]

We are considering a set of time slices corresponding to t1 ,t2,t3...t(n-1),tn [The time coordinates are in increasing order and time remains constant for each slice]
When at (t1,xA,yA,zA), I may try to visualize/pre-assess the nature of separation between the events (t(n-1),xA,yA,zA) and (tn,xB,yB,zB) along some coordinate curve[the coordinate labels are not changing]. The curve runs between the two events on two specified time slices. I know that the metrics will be changing. AS I move from (t1,xA,yA,zA) towards (t(n-1),xA,yA,zA) there may be a huge number of instants for which I may not be able to predict the nature of the curve between (t(n-1),xA,yA,zA) and (tn,xB,yB,zB)--they may turn out to be of any type timelike ,spacelike or null.

The curve has the same set of coordinate points for all predictions

We may also think in terms of parallel ensembles[I mean to say groups of time slices of the type mentioned in the first paragraph]with the surfaces corresponding to t1 as identical.We have different sets of time slices for the different sets of metric coefficients[coordinate labels are the same for the curve].And different situations are realizable -----spacelike,null or timelike connections between the same points and along the same coordinate curves .[for different ensembles]

 

[Anamitra]

I may not be able to predict the exact shape/nature of future time slices and hence the nature of separation between points lying on them[separate time slices] could be anything!

[I may have to wait a very long time for any correct prediction]

 

[Dale]

Anamitra, this is completely silly. The fact that we may be surprised about something due to ignorance on the subject is not something new to GR.

If you are ignorant of the metric in some region of spacetime then you are also ignorant of whether or not a given path through that region is timelike or spacelike. Again, a path does not change from timelike to spacelike due to the influence of gravity. If the metric is unknown then so is the nature of the path. In fact, if the metric is unknown then whether or not a given set of coordinates is even valid is also unknown.

 

[Anamitra]

Anamitra, this is completely silly. The fact that we may be surprised about something due to ignorance on the subject is not something new to GR.

If you are ignorant of the metric in some region of spacetime then you are also ignorant of whether or not a given path through that region is timelike or spacelike. Again, a path does not change from timelike to spacelike due to the influence of gravity. If the metric is unknown then so is the nature of the path. In fact, if the metric is unknown then whether or not a given set of coordinates is even valid is also unknown.

You may just think of the situation[time dependent fields depicted in #116 and #117] in contrast against stationary fields.

 

[PAllen]

We are considering a set of time slices corresponding to t1 ,t2,t3...t(n-1),tn [The time coordinates are in increasing order and time remains constant for each slice]
When at (t1,xA,yA,zA), I may try to visualize/pre-assess the nature of separation between the events (t(n-1),xA,yA,zA) and (tn,xB,yB,zB) along some coordinate curve[the coordinate labels are not changing]. The curve runs between the two events on two specified time slices. I know that the metrics will be changing. AS I move from (t1,xA,yA,zA) towards (t(n-1),xA,yA,zA) there may be a huge number of instants for which I may not be able to predict the nature of the curve between (t(n-1),xA,yA,zA) and (tn,xB,yB,zB)--they may turn out to be of any type timelike ,spacelike or null.

The curve has the same set of coordinate points for all predictions

We may also think in terms of parallel ensembles[I mean to say groups of time slices of the type mentioned in the first paragraph]with the surfaces corresponding to t1 as identical.We have different sets of time slices for the different sets of metric coefficients[coordinate labels are the same for the curve].And different situations are realizable -----spacelike,null or timelike connections between the same points and along the same coordinate curves .[for different ensembles]

Yes, you could do something like this. Note:

1) You seem to be attaching great physical significance to the coordinate labels. This is not meaningful.

2) Because of (1), you fail to see that what you describe is just because of the expression of the metric in this particular coordinate system. Please try to understand my post #105. That is how this situation would be physically experienced.

3) Take particular note that if you change coordinates to one more natural for this particular observer's world line, you wouln't observe any unusual change in coordinate properties.

 

[Anamitra]

Anamitra, this is completely silly. The fact that we may be surprised about something due to ignorance on the subject is not something new to GR.

If you are ignorant of the metric in some region of spacetime then you are also ignorant of whether or not a given path through that region is timelike or spacelike. Again, a path does not change from timelike to spacelike due to the influence of gravity. If the metric is unknown then so is the nature of the path. In fact, if the metric is unknown then whether or not a given set of coordinates is even valid is also unknown.

Anamitra, this is completely silly. The fact that we may be surprised about something due to ignorance on the subject is not something new to GR.

If you are ignorant of the metric in some region of spacetime then you are also ignorant of whether or not a given path through that region is timelike or spacelike. Again, a path does not change from timelike to spacelike due to the influence of gravity. If the metric is unknown then so is the nature of the path. In fact, if the metric is unknown then whether or not a given set of coordinates is even valid is also unknown.

Lets think of another thought experiment.

We have a timedependent metric surrounding our planet[in curved spacetime].It is known to us.Scientists on the planet have the artificial power to create a gravitational upheaval in two or more different ways . The changed metrics in case of each catastrophe are known to them. They may predict the separation between two future distant events [t1,x1,y1,z1] and [t2,x2,y2,z2] as time like or spacelike or null.


Better still we are predicting two or more types of natural upheavals in terms of gravitational changes. The metrics have been predicted for each case by the scientists. They may predict the separation between two future distant events [t1,x1,y1,z1] and [t2,x2,y2,z2] as time like ,spacelike or null according to which one occurs.

[We may apply different metrics on the same set of coordinate points]

 

[Anamitra]

I am supposed to receive some special information from a distant spacetime point after 100 years. Due to some gravitational change I get it after 10 years.If the gravitational change did not occur the points [Time after ten years,My location] and the remote spacetime point might have had a spacelike separation.If it occurs the separation might become timelike

 

[PAllen]

Lets think of another thought experiment.

We have a timedependent metric surrounding our planet[in curved spacetime].It is known to us.Scientists on the planet have the artificial power to create a gravitational upheaval in two or more different ways . The changed metrics in case of each catastrophe are known to them. They may predict the separation between two future distant events [t1,x1,y1,z1] and [t2,x2,y2,z2] as time like or spacelike or null.


Better still we are predicting two or more types of natural upheavals in terms of gravitational changes. The metrics have been predicted for each case by the scientists. They may predict the separation between two future distant events [t1,x1,y1,z1] and [t2,x2,y2,z2] as time like ,spacelike or null according to which one occurs.

[We may apply different metrics on the same set of coordinate points]

All this is over-complicated and missing key points. Coordinates have *no* meaning by themselves. They can have meaning only in conjunction with a metric expressed in them (note that the metric itself can be defined without reference to coordinates). In particular, there is no conceivable meaning to talking about different metrics on the same coordinates. If the actual physical situation is unchanged, what you mean is you've changed coordinates producing a changed expression of the metric. If the physical situation is different, then you have different coordinates *and* different metric - you just can't attach meaning to the coordinates in the abstract, separate from the metric.

 

[Dale]

Lets think of another thought experiment. ...

So what?

If you do not know the metric then you cannot calculate the interval along a path.

If you do not know a ball's mass then you cannot calculate its momentum.

If you do not know how much fuel is in the your automobile then you cannot determine how far you can travel without refueling.

If you don't know how much money is in your bank account then you cannot determine if you can afford a new computer.

Ignorance is annoying. And, yes, you can always change any solvable problem into an unsolvable one simply by reducing the number of knowns and increasing the number of unknowns. But so what?

There is nothing related specifically to GR in this particularly uninteresting line of discussion.

 

[JesseM]

We are considering a set of time slices corresponding to t1 ,t2,t3...t(n-1),tn [The time coordinates are in increasing order and time remains constant for each slice]

You're not addressing the basic issue I brought up in post #112 (and the one PAllen also discusses in post #123): how do you suppose we can define a "coordinate system" on a region of spacetime where we don't know the metric? You need to specify the details of how this is supposed to work. I did offer one suggestion earlier:

I suppose if you had a family of clocks connected by flexible springs filling space, then you could define a coordinate system where each clock had a constant position coordinate and its reading defined a time coordinate, so then you could talk about coordinates even in regions where you didn't know the metric, and define a "path" in terms of those coordinates...but if you didn't know the metric it's hard to see how you could have much basis for "expecting" that a particular path would be spacelike or timelike (except a special cases like a path of constant position coordinate, which would just be the worldline of some clock and must therefore be timelike)

 

[Anamitra]

All this is over-complicated and missing key points. Coordinates have *no* meaning by themselves. They can have meaning only in conjunction with a metric expressed in them (note that the metric itself can be defined without reference to coordinates). In particular, there is no conceivable meaning to talking about different metrics on the same coordinates. If the actual physical situation is unchanged, what you mean is you've changed coordinates producing a changed expression of the metric. If the physical situation is different, then you have different coordinates *and* different metric - you just can't attach meaning to the coordinates in the abstract, separate from the metric.

We can always set up a coordinate system in an arbitrary manner[for example we may think in terms of spherical or rectangular systems as three dimensional time-slices]. Then we can find out metrics that match against the physical aspects of the problem[this should include gravity and perhaps other factors according to the nature of the problem].

The coordinate system is of course arbitrary----it does not have to have a definite physical meaning.But once we use the physical aspects of the problem to impose the metric coefficients on them,the whole thing becomes meaningful.

This in no way serves as any impediment to my suggestions/thought experiments.

 

[Anamitra]

Preparing a coordinate system is like putting /attaching labels.You are not allowed to take off these labels at future points of time.Then you attach metric coefficients according to the physical nature of the problem. If the nature of the problem changes you simply change the metric coefficients[in a consistent way] without disturbing the labels.

 

[Anamitra]

The essential point is to have a meaningful system composed of metrics and coordinates corresponding to some physical system/situation[which includes gravity]. If the physical nature of the problem changes, you simply change the metrics without disturbing the coordinate labels.

 

[Dale]

Preparing a coordinate system is like putting /attaching labels.You are not allowed to take off these labels at future points of time.

Yes you are. That is called a coordinate transformation and you are allowed to do it as often as you like.

 

[JesseM]

Preparing a coordinate system is like putting /attaching labels.You are not allowed to take off these labels at future points of time.

And how do you "prepare a coordinate system" or "attach labels" to a region of spacetime that's so completely unknown to you that you can't even predict the metric there? What physical features of this unknown region are you attaching the labels to, so that later when you learn about what actual physical events occurred there you have a unique way of determining the coordinates of these events?

As I said before, one option would be to just assume you have an array of clocks which you use to define coordinate times and worldlines of constant coordinate position, and you know the clocks will still be in the unknown region (because they were in its past light cone) though you don't know how they'll behave. But if this is your method, you need to specify that it is since it will have some implications for your later argument...if you can think of some other method, you need to specify that. Right now it seems like you just haven't really given any thought to the problem though.

 

[Anamitra]

And how do you "prepare a coordinate system" or "attach labels" to a region of spacetime that's so completely unknown to you that you can't even predict the metric there? What physical features of this unknown region are you attaching the labels to, so that later when you learn about what actual physical events occurred there you have a unique way of determining the coordinates of these events?

As I said before, one option would be to just assume you have an array of clocks which you use to define coordinate times and worldlines of constant coordinate position, and you know the clocks will still be in the unknown region (because they were in its past light cone) though you don't know how they'll behave. But if this is your method, you need to specify that it is since it will have some implications for your later argument...if you can think of some other method, you need to specify that. Right now it seems like you just haven't really given any thought to the problem though.

The process is simple:

Metric coefficients+coordinate system----> Meaningful idea[It corresponds to some physical problem]


You choose any particular problem problem--let us call it "The Initial Problem"
Find out the metric: metric coefficients+coordinates
The above metric should correspond to the physical nature of the problem
If the physical situation changes [ex: a high density erratic mass distribution approaches the system], you simply change the metric coefficients without disturbing the coordinate labels

[Incidentally one could use "flying labels" as coordinates for a stationary system. No harm, so long as the metric coeff+coordinates combination[which we call the metric ] gives us a correct depiction of the physical situation. But calculations might become tedious requiring too much of diligence]

 

[JesseM]

Metric coefficients+coordinate system----> Meaningful idea[It corresponds to some physical problem]

But you were talking about placing coordinates in a region of spacetime where you don't yet know the metric coefficients, right? If so, nothing in your post explains how we are supposed to do that.

 

[Anamitra]

But you were talking about placing coordinates in a region of spacetime where you don't yet know the metric coefficients, right? If so, nothing in your post explains how we are supposed to do that.

You may go through post #121
You may also go through post #116

We may not always be aware of the metric ahead of us[I mean to say,the future] but we may always speculate that changes might occur----that we might get a time like,space like or null connection in future--there is a glorious uncertainty in the whole aspect of the problem

 

[PAllen]

We can always set up a coordinate system in an arbitrary manner[for example we may think in terms of spherical or rectangular systems as three dimensional time-slices]. Then we can find out metrics that match against the physical aspects of the problem[this should include gravity and perhaps other factors according to the nature of the problem].

The coordinate system is of course arbitrary----it does not have to have a definite physical meaning.But once we use the physical aspects of the problem to impose the metric coefficients on them,the whole thing becomes meaningful.

This in no way serves as any impediment to my suggestions/thought experiments.

JesseM has suggested several times you think about what it means to set up a coordinate system. I doubt I can do better, but I'll try again.

Suppose you want to label an event B 3 units in x direction from event A (events are points in space time; they have no history - they are specific events somewhere, sometime, in the history of the universe). This labeling has no meaning at all until you know the metric and can express it in terms of x and other labels. Depending on how you do this, 3 in x direction can mean 3 hours later on a clock, 3 kilometers east, 3 degrees counterclockwise, whatever. It is only the metric that gives x any meaning at all. If the metric says x direction is timelike, than x has the character of time for some clock; if the metric says it is spacelike, then it is distance for some path of simultaneity.

More naturally, you can set up coordinates by (perhaps idealized) measurements. Then the measurements determing the nature of of the coordinates. Measurements, of course, take full account of the metric. If you define x by a mechanism for measuring distance, it will represent distance no matter where or when in the universe you do it, no matter what the gravitational field.

If you are 'thinking' about the the interval from (t,x)=(5,5) to (5,7), where 5 is in the future, and you have don't know the metric for this region of space *time*, and don't define any measurement you will do,, then you cannot have any expectation of what they mean. I cannot fathom what you mean by 'expecting' a meaning for this separately from a measurement procedure or defining the metric.

Note, if you define this, for example, by saying that when my watch says 3, I will define my spacetime position to be (3,5), then I will send out a rader signal and if I get it back when my watch says 7, (still calling my postion x=5), then the event of its bouncing off something I will label (5,7). Using such a procedure you would know, at all times, and all gravity situations, that you would be defining a spacelike interval between (5,5) and (5,7), and you could call them 2 lightseconds apart (thus calling your watch units seconds, and x unit lightseconds). An observer elsewhere in the universe might disagree radically on how far apart these events were, but they would certainly agree the separation between them was spacelike.

 

[JesseM]

You may go through post #121
You may also go through post #116

Neither post contains any information about what physical procedure we are supposed to use to attach coordinate labels to a region of spacetime where we don't know the metric. Can you please just give a specific answer to this question?

 

[PAllen]

If the physical situation changes [ex: a high density erratic mass distribution approaches the system], you simply change the metric coefficients without disturbing the coordinate labels

Maybe this is the core confusion. What are these labels attached to while you are changing the metric? If they are attached to physical events and measurements, the metric is discovered by these measurements, not the other way around. If they are not, then the only meaning they have is determined by the way the metric is expressed in terms of these labels. Depending on how you change the metric, you may redefine x as time on a clock, distance along a ruler, or angle.

 

[PAllen]

Maybe I see a way to define what Anamitra is getting at in a sensible way. Suppose you define 'reasanoble coordinate' for observing what you can see in the universe. Specifically, suppose you choose to extend Fermi Normal cordinates as far in distance and time as you reasonably can. You find that out to 10 lightyears, there are no signficant deviations from Lorentz geometry except near your sun and planets. You think you can extend them well into the future because you haven't detected anything that would make this invalid. Then an isolated black hole passes a few light years away. Now you find that doing your best to extend these coordinates (still keeping them Fermi Normal based on your world line as the time axis), you can't avoid varying coordinate speed of light near the black hole; necessarily, this also means that a coordinate path defined e.g. by: ( x - x0) = .5 (t - t0) that is is timelike for t0 before the arrival of the black hole, and for any x0 far from the black hole, now describes a spacelike curve for some x0 and t0.

Of course the only sense in which you can say this is unexpected is that you previously lacked knowledge of the approaching black hole (or you didn't know about GR). So it is strange to call this unexpected. Further, there is, of course, no spactime path that changed nature because of the approaching black hole. The best you could say is that for t0 in your future you made an erroneous guess about the corresponding metric. It was your guess that got corrected, not any spacetime path changing nature.

 

[Anamitra]

When we frame the metrics for a stationary field [eg:Schwarzschild Geometry] we extend the time axis into the distant future expecting nature to be kind towards us[maintaining the stationary nature].Then we have the metrics for the stationary field.In case something happens[some gravitational upheaval] we can maintain our coordinate grid and change the nature of the metrics.We may also think of changed metrics for the future speculating different types of gravitational changes.
This was of course implied in the previous posts

 

[JesseM]

When we frame the metrics for a stationary field [eg:Schwarzschild Geometry] we extend the time axis into the distant future expecting nature to be kind towards us.

You can only extend it under the assumption that the metric will be stationary (or some other assumption about the metric). If you make this assumption and then say "I wonder whether some point at a future time T and position X will lie on a timelike path from my current position", but then the metric changes contrary to your assumption, how are you supposed to decide what physical event in spacetime actually has coordinates T and X? The labels T and X simply become meaningless if your basis for them was the assumption that the metric would stay stationary when in fact it didn't, you have to construct a new coordinate system if you want to label events in the region of spacetime that didn't match your expectations.

 

[Anamitra]

You can only extend it under the assumption that the metric will be stationary (or some other assumption about the metric). If you make this assumption and then say "I wonder whether some point at a future time T and position X will lie on a timelike path from my current position", but then the metric changes contrary to your assumption, how are you supposed to decide what physical event in spacetime actually has coordinates T and X? The labels T and X simply become meaningless if your basis for them was the assumption that the metric would stay stationary when in fact it didn't, you have to construct a new coordinate system if you want to label events in the region of spacetime that didn't match your expectations.

Actually I am constructing a new metric with the old coordinate grid[t,x,y,z] and new metric coefficients.