Can different paths in spacetime have the same separation?

[Anamitra]

If a person on a planet wants to receive information from a distant source,at a spacelike separation, he has to wait [ to advance his time coordinate]till he can connect the distant point by a null geodesic.May be he has to wait for hundred years or more to receive some particular type of information/special information emitted at a particular instant of time from the distant source. But if the spacelike path gets converted into a timelike or a null path by gravitational influences he gets the advantage of an earlier reception.

The spacelike path may undergo bit by bit transformation into timelike or null segments and then get reconverted [into space like segments] as the signal/information passes on!

[We are considering the influences of a time dependent field here]

 

[Anamitra]

We consider the metric:
$${ds}^{2}{=}{{g}_{00}}{dt}^{2}{-}{{{g}_{11}}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{3}$$

In a time dependent field the values of the metric coefficients could change with time yielding positive ,negative [or zero] sign of $${ds}^{2}$$. This idea is apparent in the past few posts.
In #34 the scattering experiment may be exemplified by a pair creation taking place at the tip of the cone

 

[Anamitra]

The Light Cone in General Relativity

For the surface of the cone[the generating lines]:

$${\frac{physical distance}{physical time}{=}{c}$$
C=1 in the natural system ofunits.
This implies that
$${\frac{coordinate distance}{coordinate time}\neq}{c}$$
In the general case[for the same lines].

Let us think of a tip of a typical cone[coordinate cone] lying on the x-y plane. The time axis is perpendicular to this plane. The cone is being considered in relation to the coordinate values and not the physical values.If we draw a circle[on the xy plane] with the tip of the cone as center and some fixed radius[representing coordinate value] ,the radii in different directions will represent unequal physical distances. [The physical times will also have different values for the same coordinate value of time].The shape of the coordinate cone will be a distorted one since the lines emanating from the tip along the surface will not necessarily make 45 degrees with each radial line in the xy plane. They will make different angles in different directions .The coordinate cone will be a distorted one.Straight lines joining the tip to the interior points will represent time like separations,those joining the tip to the surface null separations and the others space separations Since events are labeled with coordinate values and not with physical values it is important to consider the coordinate cone. With changes in the values of the metric coefficients the distortion in the surface would undergo further changes.Points outside the cone may become points inside it and vice-versa.
[ The surface of the cone itself would move[undergo distortions] and pass across different points when the metric coefficients change in value giving an impression as if the points have crossed over]
That could be the possible light cone mechanism of interconversion of spacelike and time like intervals[in the rectangular system]

[A time dependent field has been considered in the last part of the discussion]

 

[PAllen]

I have been thinking of a simple example[of course one including a sharp bend]. We take two points inside the light cone.These points should have the same temporal coordinates.If a signal were to pass between them along a curve lying on t=const surface it would be definitely be an infinitely fast signal[traveling across a spacelike interval].
But we could think of a broken line running to the tip of the cone from the first point and then up to the second one along a straight line[A timelike interval].

The first motion is along the negative direction of time.Antiparticles can of course do that. Basically we have a scattering experiment going on at the tip of the cone!

One could try adding gravity to the whole situation/similar situations to convert a spacelike interval ito a time like one or vice-versa[by considering changes in the metric coefficients in a time dependent field]

That is exactly what is meant to be precluded by the requirement of smoothness. Any way of smoothing the bend introduces a spacelike section making the path mixed. Even with the sharp bend, you can't call the whole curve timelike because ds^2 is undefined at the bend. Thus, a pure timelike curve needs to be defined as one where ds^2 is everywhere defined and continuous and positive (or negative, depending on your signature convention). Similarly for a pure spacelike path.

 

[PAllen]

One could always consider gravity as an agent that could work out an interconversion between spacelike and timelike intervals. A person moves across a timelike interval and then due to a change in the values of the metric coefficients the interconversion takes place.He has in his memory the old coordinates. How should he feel like now?

A coordinate transformation can never change the causality relation between events. Further, even in the most extreme geometry, the classification of a path (timelike, spacelike, or mixed) is invariant with respect to coordinate changes. What is possible in extreme geometry is that the causal relations between events are ill defined: you can have two events such that along one timelike path P1 is after P0, while along a different timelike curve, P0 is after P1 (keeping the coordinate system the same for this comparison; i.e. not just replacing t with -t). However, if this situation is true, it is intrinsic and not coordinate dependent.

 

[Anamitra]

A coordinate transformation can never change the causality relation between events. Further, even in the most extreme geometry, the classification of a path (timelike, spacelike, or mixed) is invariant with respect to coordinate changes. What is possible in extreme geometry is that the causal relations between events are ill defined: you can have two events such that along one timelike path P1 is after P0, while along a different timelike curve, P0 is after P1 (keeping the coordinate system the same for this comparison; i.e. not just replacing t with -t). However, if this situation is true, it is intrinsic and not coordinate dependent.

If the very interval changes in one system[due to the effect of gravity] it should change in other coordinate systems.After the change the intervals should again become equal in all systems.Initially we had an invariant set.
Finally we have another invariant set.

 

[PAllen]

We consider the metric:
$${ds}^{2}{=}{{g}_{00}}{dt}^{2}{-}{{{g}_{11}}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{3}$$

In a time dependent field the values of the metric coefficients could change with time yielding positive ,negative [or zero] sign of $${ds}^{2}$$. This idea is apparent in the past few posts.
In #34 the scattering experiment may be exemplified by a pair creation taking place at the tip of the cone

Note that this doesn't change the statement that the causal relation between two given events is invariant, and that the classification of a path into timelike, spacelike, or mixed is invariant. You would instead have the feature that e.g. (t,x,y,z) = (0,0,0,0) and (0,1,0,0) are spacelike in causal relations (no timelike or null path exists between them), while two *different events*, e.g. (3,3,3,3) and (3,4,3,3) have timelike relation between them. Your similar paths at different value of the t coordinate are completely different paths, different in fundamental nature.

Note also that there are some limits on the metric functions you can specify GR. You must everywhere have locally Minkowski geometry. Further, the equations of GR must actually be satisfied. Arbitrarily chosen metric functions can easily fail to meet these conditions.

 

[PAllen]

If the very interval changes in one system[due to the effect of gravity] it should change in other coordinate systems.After the change the intervals should again become equal in all systems.


Initially we had an invariant set.
Finally we have another invariant set.

A solution in GR is a space*time* geometry. The statement 'gravity causes the geometry to change later' has no meaning.

 

[Anamitra]

Note that this doesn't change the statement that the causal relation between two given events is invariant, and that the classification of a path into timelike, spacelike, or mixed is invariant. You would instead have the feature that e.g. (t,x,y,z) = (0,0,0,0) and (0,1,0,0) are spacelike in causal relations (no timelike or null path exists between them), while two *different events*, e.g. (3,3,3,3) and (3,4,3,3) have timelike relation between them. Your similar paths at different value of the t coordinate are completely different paths, different in fundamental nature.

The events (3,3,3,3) and (3,4,3,3) will never correspond to a time like separation since dt=0
Physical separation=$${g}_{00}{dt}{=}{0}$$

This holds for all values of $${g}_{00}$$ including the changed values.

 

[Anamitra]

A solution in GR is a space*time* geometry. The statement 'gravity causes the geometry to change later' has no meaning.

Invariance is related to the transformation of the coordinate systems. This does not impose any restriction on the interval itself changing to a new invariant value wrt all systems

 

[PAllen]

The events (3,3,3,3) and (3,4,3,3) will never correspond to a time like separation since dt=0
Physical separation=$${g}_{00}{dt}{=}{0}$$

This holds for all values of $${g}_{00}$$ including the changed values.

If one of the other metric coefficients changes sign, this could happen. Timelike simply means ds^2 is positive (using your convention). Which coordinate is 'most timelike' can, in principle, vary from one region of spacetime to another. Generally, one would try to avoid such coordinate systems, but they are not illegal and in extreme geometries it could be hard to avoid coordinate systems with unusual properties.

 

[Dale]

One could try adding gravity to the whole situation/similar situations to convert a spacelike interval ito a time like one or vice-versa[by considering changes in the metric coefficients in a time dependent field]

One could always consider gravity as an agent that could work out an interconversion between spacelike and timelike intervals.

The spacelike path may undergo bit by bit transformation into timelike or null segments and then get reconverted [into space like segments] as the signal/information passes on!

If the very interval changes in one system[due to the effect of gravity] it should change in other coordinate systems.After the change the intervals should again become equal in all systems.

Wow, that is like a shotgun barrage of posts. There is a common running theme in the last several posts that indicates a misunderstanding. Gravity cannot change a timelike interval into a spacelike interval. You can have coordinate systems in which a given coordinate basis vector changes smoothly from timelike to spacelike even in flat spacetime, but the interval along a given path is always timelike or spacelike. Gravity cannot change that.

We consider the metric:
$${ds}^{2}{=}{{g}_{00}}{dt}^{2}{-}{{{g}_{11}}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{3}$$

In a time dependent field the values of the metric coefficients could change with time yielding positive ,negative [or zero] sign of $${ds}^{2}$$. This idea is apparent in the past few posts.

I think this may be the source of your confusion. Let's specify two paths
path A: (T1,x,0,0) where X1<x<X2
path B: (T2,x,0,0) where X1<x<X2
Now,if the metric is static then A and B will have the same spacetime interval, but if the metric is not then it is possible that A may be spacelike while B is timelike. But A and B are different paths. The fact that B is timelike at T2 does not in any way imply that A is timelike at T1, and A simply does not exist at T2.

 

[Anamitra]

I think this may be the source of your confusion. Let's specify two paths
path A: (T1,x,0,0) where X1<x<X2
path B: (T2,x,0,0) where X1<x<X2
Now,if the metric is static then A and B will have the same spacetime interval, but if the metric is not then it is possible that A may be spacelike while B is timelike. But A and B are different paths. The fact that B is timelike at T2 does not in any way imply that A is timelike at T1, and A simply does not exist at T2.

If T2 is a constant quantity coordinate separation [wrt time ] is zero
dt=0
Physical separation of time = $${g}_{00}{dt}{=}{0}$$

This holds for all changes of $${g}_{00}$$

Path B should continue to remain spacelike.[Integration should yield a zero result if dt=0 at all points of the curve]

Are you assuming unusual metrics or something like that like PAllen?[#46]

 

[Dale]

If T2 is a constant quantity coordinate separation [wrt time ] is zero
dt=0

Yes.

Physical separation of time = $${g}_{00}{dt}{=}{0}$$

This holds for all changes of $${g}_{00}$$

I don't like your term "physical time". It is not used by anyone other than yourself. Please don't bring it into this discussion.

Path B should continue to remain spacelike.[Integration should yield a zero result if dt=0 at all points of the curve]

Are you assuming unusual metrics or something like that like PAllen?[#46]

I am considering metrics of the form you suggested in post 37. In such metrics just because t is timelike at T1 does not imply that it is timelike at T2.

 

[Anamitra]

We consider the metric:
$${ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}$$

if dt=0
we have,
$${ds}^{2}{=}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}$$

$${ds}^{2}$$<0 that is, the interval is spacelike unless one of the coefficients($${g}_{11}$$ or $${g}_{22}$$ or $${g}_{11}$$) or some of them/all of them are negative. And that is quite unusual.

 

[Dale]

Don't forget that per your post 37 the components of the metric are functions of t. So any of them may be positive at T1 and negative at T2. This is, in fact, required for what you were considering through your barrage of posts last night.

 

[Anamitra]

If we consider the speed of a particle/point along our path [in consideration]it becomes infinitely large--both the coordinate speed and the physical speed. This is in view of the fact dt=0. This is valid even if the coefficients g(ii) take on unusual negative values.How does the path remain timelike in such a situation?

We may consider a limiting process speed -->an infinitely large quantity as dt--->0

One may avoid the problem by assuming [rather by imposing the condition]
$${-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}$$ is not positive.[at least for a time like path with dt=0 in the background]

If $${-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}$$ is positive for a timelike path with dt=0 both timelike and spacelike paths will allow faster than light signals!

 

[PAllen]

If we consider the speed of a particle/point along our path [in consideration]it becomes infinitely large--both the coordinate speed and the physical speed. This is in view of the fact dt=0. This is valid even if the coefficients g(ii) take on unusual negative values.How does the path remain timelike in such a situation?

We may consider a limiting process speed -->an infinitely large quantity as dt--->0

One may avoid the problem by assuming [rather by imposing the condition]
$${-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}$$ is not positive.[at least for a time like path with dt=0 in the background]

If $${-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}$$ is positive for a timelike path with dt=0 both timelike and spacelike paths will allow faster than light signals!

The 4 velocity along a time like path is differentiation with respect to proper time (tau), and is always a unit vector. There is no such thing as velocity along a spacelike path. Differentiation with respect to coordinate t along a timelike path with constant value t is meaningless and undefined.

 

[Anamitra]

The 4 velocity along a time like path is differentiation with respect to proper time (tau), and is always a unit vector. There is no such thing as velocity along a spacelike path. Differentiation with respect to coordinate t along a timelike path with constant value t is meaningless and undefined.

Four velocity along a time like path is well defined and it conforms to a standard definition.How do you calculate it for the special timelike path under consideration for which dt=0? I mean to sayhow do you get the proper time interval to carry out the differentiation?

Possibly you mean to establish the idea that we may have exceptions for timelike paths for which the four velocity cannot be calculated.

 

[JesseM]

How do you calculate it for the special timelike path under consideration for which dt=0?

Why do you say this path is "timelike"? In most coordinate systems typically used in physics (Minkowski coordinates, for example), if dt=0 along a path then the path is spacelike, not timelike.

 

[Anamitra]

We are analyzing a particular type of Time Like curve DaleSpam has tried to illustrate in #48[Path B]
Subsequent posts are concerned with such paths--Jesse should consider such special type of timelike paths in order to maintain the relevance of the discussion.
[You are requested to go through the subsequent posts[#48 and the following ones]

 

[PAllen]

Four velocity along a time like path is well defined and it conforms to a standard definition.How do you calculate it for the special timelike path under consideration for which dt=0? I mean to sayhow do you get the proper time interval to carry out the differentiation?

Possibly you mean to establish the idea that we may have exceptions for timelike paths for which the four velocity cannot be calculated.

Nothing special at all. d tau is based on the metric as you've specified; compute dt / d tau, dx / d tau, etc. All perfectly well defined for curve where d tau is positive (as it is along the curve under discussion).

 

[PAllen]

Why do you say this path is "timelike"? In most coordinate systems typically used in physics (Minkowski coordinates, for example), if dt=0 along a path then the path is spacelike, not timelike.

This discussion started out with Anamitra proposing that the metric components could be functions including coordinate time, so that a 'similar' coordinate path could be spacelike in one region and timelike in another. Dalespam and I simplified to the case where you could have dt=0 along a path, yet it could, indeed, be spacelike in one region and timelike in another. The only thing 'special' about this situation is that it means that which coordinate(s) have time like character is different in different regions of spacetime. Perhaps if you read over the whole thread (mostly ignoring one of my misunderstandings, which Dalespam clarified), focusing on what Dalespam has said, you could clarify for Anamitra better than I've done so far.

 

[Anamitra]

Nothing special at all. d tau is based on the metric as you've specified; compute dt / d tau, dx / d tau, etc. All perfectly well defined for curve where d tau is positive (as it is along the curve under discussion).

The time component of the four velocity[As suggested by PAllen] is zero. Further differentiation wrt to propertime[ds] yields time component of the momentum vector[multiplication by rest mass is required] which is again zero. It is zero energy particle![Assuming a particle is capable of moving along a timelike curve/path]

 

[JesseM]

We are analyzing a particular type of Time Like curve DaleSpam has tried to illustrate in #48[Path B]
Subsequent posts are concerned with such paths--Jesse should consider such special type of timelike paths in order to maintain the relevance of the discussion.
[You are requested to go through the subsequent posts[#48 and the following ones]

Well, DaleSpam didn't specify any particular metric, but if you had one, then you could use it to compute proper time along the timelike path in the standard way, integrating $$\sqrt{g_{tt} dt^2 + g_{xx} dx^2 + g_{yy} dy^2 + g_{zz} dz^2 }$$ along the path. In Dalespam's example only the x-coordinate varies so dt=dy=dz=0 along the path, meaning if the path varies from X1 to X2 and the path is timelike, you can calculate the proper time using the integral $$\int_{X1}^{X2} \sqrt{g_{xx} } \, dx$$. Does this answer your question "I mean to sayhow do you get the proper time interval to carry out the differentiation?"

 

[Dale]

If we consider the speed of a particle/point along our path [in consideration]it becomes infinitely large

If a path is spacelike then it cannot represent the worldline of a particle. If a path is timelike then the velocity wrt any other (local) timelike path will be < c.

 

[Anamitra]

Nothing special at all. d tau is based on the metric as you've specified; compute dt / d tau, dx / d tau, etc. All perfectly well defined for curve where d tau is positive (as it is along the curve under discussion).

For the Special Time Like path ,refered to by PAllen in the qouted section,the coordinate interval [time]between a pair of events is zero[since dt=0 for each and every subsection of the path].The proper time for such a path is the length of the path itself[ds=proper time interval]. If a person moves along this path with a clock in his hand what is he going to observe, considering the fact that dt=0 for each and every subsection of the path?
[A person should be capable of moving along a timelike path. Incidentally PAllen and DaleSpam are claiming the existence of a TIMELIKE PATH for which dt=0 for each infinitesimal subsection [ds^2>0 according to my convention].]

 

[JesseM]

For the Special Time Like path the coordinate interval [time]between a pair of events is zero[since dt=0 for each and every subsection of the path].The proper time for such a path is the length of the path itself[ds=proper time interval]. If a person moves along this path with a clock in his hand what is he going to observe, considering the fact that dt=0 for each and every subsection of the path?

Clocks measure proper time, not coordinate time. Different points on the path have different values of proper time, so he will see his clock tick forward as he moves along the path. In some coordinate system the worldline you are moving along right now has a constant t-coordinate, do you notice anything strange happening?

 

[Anamitra]

Clocks measure proper time, not coordinate time. Different points on the path have different values of proper time, so he will see his clock tick forward as he moves along the path. In some coordinate system the worldline you are moving along right now has a constant t-coordinate, do you notice anything strange happening?

Just tell me the value of propertime for the aforesaid path [between a pair of events on it]?Is it going to conform to the physical notion of the time interval as the person travels along the path between the events with his own clock?
[You should be careful enough to give due consideration to the nature of the very special type of the timelike of path we are dealing with--dt=0 for each and every infinitesimal subsection]
[The existence of such a path has been suggested by PAllen and DaleSpam]

 

[JesseM]

Just tell me the value of propertime for the aforesaid path [between a pair of events on it]?

I did that. Did you read post #60?

Is it going to conform to the physical notion of the time interval as the person travels along the path between the events with his own clock?

Yes, of course. That's the physical meaning of "proper time"--it always corresponds to clock time, and "clock time"/"proper time" are both coordinate-independent notions (of course for a given spacetime geometry, the equations for the components of the metric change depending on what coordinate system you use in that spacetime, but when you integrate $$\sqrt{g_{tt} dt^2 + g_{xx} dx^2 + g_{yy} dy^2 + g_{zz} dz^2 }$$ along a path using the correct form of the metric tailored to the coordinate system which you're using to describe the path, you will get the same answer regardless of what coordinate system you choose).

[You should be careful enough to give due consideration to the nature of the very special type of the timelike of path we are dealing with--dt=0 for each and every infinitesimal subsection]
[The existence of such a path has been suggested by PAllen and DaleSpam]

Coordinate time is completely irrelevant, only proper time matters if you want to know what physical clocks are doing. If you think it somehow makes a difference that dt=0, I think you're misunderstanding something basic about the physical meaning of proper time (and the fact that it's independent of what coordinate system you choose--we could easily describe the exact same physical path in a different coordinate system where dt was not zero, and the proper time along the path would necessarily be exactly the same).

 

[Anamitra]

Well, DaleSpam didn't specify any particular metric, but if you had one, then you could use it to compute proper time along the timelike path in the standard way, integrating $$\sqrt{g_{tt} dt^2 + g_{xx} dx^2 + g_{yy} dy^2 + g_{zz} dz^2 }$$ along the path. In Dalespam's example only the x-coordinate varies so dt=dy=dz=0 along the path, meaning if the path varies from X1 to X2 and the path is timelike, you can calculate the proper time using the integral $$\int_{X1}^{X2} \sqrt{g_{xx} } \, dx$$. Does this answer your question "I mean to sayhow do you get the proper time interval to carry out the differentiation?"

The method prescribed in the quoted text indicates that the proper speed is equal to the speed of light.What speed should be observed by an observer standing on the ground say at the initial point of the motion?What time of travel should the stationary observer [say one at the initial point] calculate/observe?

 

[JesseM]

The method prescribed in the quoted text indicates that the proper speed is equal to the speed of light.

How are you defining "proper speed"? The method given in my quote dealt only with calculating proper time, not "proper speed". Proper velocity is defined as the rate that coordinate position is changing relative to proper time (as opposed to coordinate time as with coordinate velocity), but it can take any value from 0 to infinity, so if you think proper speed would be "equal to the speed of light" I guess you have invented your own definition?

What speed should be observed by an observer standing on the ground say at the initial point of the motion?What time of travel should the stationary observer [say one at the initial point] calculate/observe?

Are you asking about coordinate time and speed in some coordinate system? (If so, which one? Is it the inertial rest frame of the observer, or is it some alternate coordinate system like the one DaleSpam discussed where a timelike worldline may have dt=0? If dt=0 for a journey in some coordinate system, then of course in that coordinate system the coordinate time of travel is zero and the coordinate speed is infinite)

 

[Anamitra]

The observer is at rest at the initial point of motion.He is trying to investigate the motion from a laboratory.

1) What formula should he use to calculate the speed or to define the speed of the particle at some point of the path?
2) What formula should he use to calculate the time interval or to define the time interval of the particle for the entire motion?

 

[Anamitra]

I would request the audience to consider the following link in regard of the general nature of the ongoing conversation::

https://www.physicsforums.com/showpost.php?p=3062372&postcount=59

 

[Dale]

Anamitra, given some timelike path, x, parameterized by a variable, lambda, the proper time along that path is defined* as:
$$\tau = \int \sqrt{-\frac{1}{c^2}\left(g_{\mu\nu} \frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} \right)} d\lambda$$

Since by definition the interval squared is always negative for a timelike interval, this value is always strictly positive, regardless of whether or not the coordinates are unusual.

*using the usual convention where spacelike intervals squared are positive and have units of length squared.