On the Nature of ds[in GR]

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We consider an spacelike infinitesimal separation [tex]{ds}^{2}[/tex]<0

ds=+ or -ib [an imaginary quantity]

Now I integrate ds along some path from A to B. What happens if the imaginary parts cancel out on integration[if we can manage to cancel them out]?I mean, is it physically significant in any way?

We may have a slight variation of the problem:

Three spacetime points,A B and C lying at the corners of an infinitesimally small triangle are chosen[Of course this triangle does not lie on a flat surface]

ds from A to B=ib
ds from B to C =-ib




ds from A to C via B=0

Now is it possible to locate a direct path from A to C[which is not through C] which gives a null separation?

Is it quite possible that ds may simultaneously correspond to the two types of separation along different paths.

[The paths connecting the points are not straight lines[in general] but are infinitesimally small in length]
 
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Is it possible to have the separation between a pair of points[fixed points] at a finite distance as spacelike ,time like or null along different paths if the metric coefficients along the different paths have suitable values catering to our requirement?
 

tiny-tim

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Hi Anamitra! :smile:

(try using the X2 icon just above the Reply box :wink:)
ds=+ or -ib [an imaginary quantity]

Three spacetime points,A B and C lying at the corners of an infinitesimally small triangle are chosen[Of course this triangle does not lie on a flat surface]

ds from A to B=ib
ds from B to C =-ib
whether you choose + or - is entirely arbitrary …

you have a parameter s, which you are free to have increase in either direction …

when ds2 is real, a line AB can have s increasing from A to B, or from B to A (obviously, when a change in t is involved, we prefer the sign that agrees with the sign of chamge of t) …

if the metric is +---, and A and B are (0,0,0,0) and (0,0,0,1), then ds2 = -1, but A and B are simultaneous, so it doesn't really matter which way your parameter goes, you can choose s to be increasing imaginary or decreasing imaginary

if C is (.1,.1,0,0), then AB and BC both have ds2 < 0, but AC is null. :wink:
Is it possible to have the separation between a pair of points[fixed points] at a finite distance as spacelike ,time like or null along different paths if the metric coefficients along the different paths have suitable values catering to our requirement?
Not without some topological weirdness like a wormhole.
 
In curved spacetime the the four-dimensional distance between a pair of points at a finite distance depends on the path of integration.

In the infinitesimal sense ds^2 depends only on the pair of points concerned and not on the path connecting them[nevertheless one may think of several infinitesimally short paths connecting the points.]

In a very small region of space we are inclined to approximate curved space-time with flat space-time.But different paths emanating from a point[4D point] may have drastically different values[for curved spacetime] of the metric coefficients in different directions even in an infinitesimally small area surrounding the point..But we use ds^2 as a the function of the point-pair without any regard for path.What impact does this have/should it have on the physics of curved spacetime?

[The value of ds^2 as we get from its definition may not hold for all infinitesimally short paths connecting the two close points--perhaps we could find one or two/or a few of them for which the value matches!]
 
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When we integrate ds along a specified path between a pair of space-time points ,the exact nature of the path[its equation] is known to us even for the smallest sections [infinitesimal sections] of the route.

Here we are considering the path even for the infinitesimal portions.

In view of the above is it reasonable to consider ds^2 as a path independent quantity---something which depends only on the pair of points[their coordinates]?

For a pair of infinitesimally close points in curved spacetime with a fixed value of ds^2[positive ,negative or zero] there is a possibility that we could connect them with infinitesimally short curves of different nature--spacelike,timelike or null[by considering the exact equations of the connecting curves and carrying out the integrations along the concerned curves ]

This is due to the fact that the metric coefficients may show a considerable amount of variation in a small region of space[as an illustration we may consider different directions showing considerable variations in the values of such coefficients in the general case]
 
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tiny-tim

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In view of the above is it reasonable to consider ds^2 as a path independent quantity---something which depends only on the pair of points[their coordinates]?
I'm not following you :confused:

even on an ordinary 2D piece of paper, the distance between two points obviously depends on the path.
 
Lets draw a space time diagram pertaining to flat spacetime on a piece of paper. We a consider a pair of points and connect them with several world lines.

Next Step:Motion in flat space time is considered in the x-y plane with the time axis perpendicular to the x-y plane.Again we consider a pair of events and connect them with different world lines. The nature of [tex]{\int {ds}}[/tex] in respect of sign should remain unchanged.

Now let us replace flat spacetime by curved spacetime.The spatial distances and the temporal separations will be different for the different curves.

So far we have considered spatial distances with greater focus. Now let us pass on to spacetime distances.

First we consider a pair of points outside the light cone[in flat spacetime]. We get spacelike separation for all paths connecting them[ds^2<0 in my convention]. You should not get anything else.

But the situation is quite different with general relativity ie in relation to curved spacetime considerations which can allow a great amount of diversity in the values of the metric coefficients.For a pair of points we may,in the general case, have three types of routes[world lines]--spacelike,timelike or null.This I have already explained in the previous posts.
 
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The important aspect to consider is the nature of the integral [tex]{{\int {ds}}[/tex] in respect of its sign and not spatial distances

The sign of the above integral [for different paths] is not supposed to change for different paths so far as flat spacetime is concerned.But with curved space-time there is a certain amount of possibility[of the sign changing] in view of the immense variety/complexity General Relativity has to offer.
 
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PAllen

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The important aspect to consider is the nature of the integral [tex]{{\int {ds}}[/tex] in respect of its sign and not spatial distances

The sign of the above integral [for different paths] is not supposed to change for different paths so far as flat spacetime is concerned.But with curved space-time there is a certain amount of possibility[of the sign changing] in view of the immense variety/complexity General Relativity has to offer.
I've seen different author's take different positions on this. What is clear is that the physically meaningful cases are where ds is everywhere timelike, everywhere null, or everywhere spacelike along the path. These are respectively, a possible world line (computing a proper time), a possible light path, or a possible line of simultaneity along which you are computing a proper length. Some authors declare interval along a mixed path undefined; some say you take absolute value of the ds**2, but mixed is not physically meaningful. You can also compute a complex interval, if you want, as you suggest. If you choose complex (e.g. real part being aggregate proper time, imaginary being aggregate proper length), I don't think your scenario of canceling imaginary contributions can occur; you always would take principal square root of the negative number, so all imaginary contributions would be positive imaginary; all real contributions would be positive real.
 
Actually one does not need to cancel imaginary parts along a curve.I can choose different paths connecting the pair of points[space-time]. Buy suitable choice of the values of the metrics the paths may be separately[even for infinitesimal segments] spacelike ,timelike or null.So in the general case we may have three different types of paths[even in respect of their infinitesimal subsections] connecting the two points.
 
A space like interval may be connected by an infinitesimal null curve or a timelike curve in curved spacetime[we are considering the general case]. I have tried to bring out this point in # 4 and # 5.

A possibility is there--that is exactly what I want to say.
 

PAllen

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Actually one does not need to cancel imaginary parts along a curve.I can choose different paths connecting the pair of points[space-time]. Buy suitable choice of the values of the metrics the paths may be separately[even for infinitesimal segments] spacelike ,timelike or null.So in the general case we may have three different types of paths[even in respect of their infinitesimal subsections] connecting the two points.
Along a given path, with a given geometry, the nature of the path is invariant with respect to any coordinates: either all timelike, all null, all spacelike, or mixed; if mixed, which parts of the path have which character is invariant.

Between two chosen events, only one type of non-mixed path is possible. If one event within the lightcone of the other, you can have pure timelike or mixed paths between them; there will never be pure spacelike or null paths. If one event is outside the light cone of the other, then all pure paths between them will be spacelike; otherwise, all pure paths between them will be null.

You asked, in your OP:

"What happens if the imaginary parts cancel out on integration[if we can manage to cancel them out]?"

That is what I was answering with my statement that this is impossible. If you are choosing to define complex invariant interval, all imaginary contributions will add, and all real contributions will add.
 
One may consider a pair of points on the surface of the light cone itself. We may connect them by null ,spacelike or timelike paths.As the partic/point under obsrrvation moves the tip of the cone moves with it. In general relativity[curved spacetime] the cone itself can have different vertical angles at different points and the axis can also tilt[considering coordinate values].With the movement of the cone we may choose a curvilinear track along its surface.At each step we consider three types of infinitesimal paths.So we can definitely have three types of pure curves connecting the events.

One may consider a point on the tip of the cone and another in the interior. We can consider a path on the surface of the cone to a very small distance.Once the tip gets there the cone may tilt and the surface may angle towards the final point
 
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But with curved space-time there is a certain amount of possibility[of the sign changing] in view of the immense variety/complexity General Relativity has to offer.
This is correct. In fact, a closed timelike curve relies on this possibilitiy. It is closed, so the trivial path is null, but there is also a timelike path connecting the event to itself.
 

PAllen

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One may consider a pair of points on the light cone itself. We may connect them by null ,spacelike or timelike paths.As the particle moves the tip of the cone moves with it. In general relativity[curved spacetime] the cone itself can have different vertical angles at different points and the axis can also tilt[considering coordinate values].With the movement of the cone we may choose a curvilinear track along its surface.At each step we consider three types of infinitesimal paths.So we can definitely have three types of pure curves connecting the events.

One may consider a point on the tip of the cone and another in the interior. We can consider a path on the surface of the cone to a very small distance.Once the tip gets there the cone may tilt and the surface may angle towards the final point
This is incorrect. Between two chosen events, only one type of pure path is possible. One event is either inside the lightcone(s) of the other, outside, or on them. This fact is invariant and determines the only type of pure paths between them.
 
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Between two chosen events, only one type of non-mixed path is possible. If one event within the lightcone of the other, you can have pure timelike or mixed paths between them; there will never be pure spacelike or null paths. If one event is outside the light cone of the other, then all pure paths between them will be spacelike; otherwise, all pure paths between them will be null.
Consider a muon at rest at the origin of an inertial frame in flat spacetime. The worldline of the muon is a timelike curve connecting the creation and decay events. Let the lifetime of the muon be T, and consider the family of curves with 2 straight segments connecting the creation (0,0,0,0) and decay (cT,0,0,0) events through an event at (cT/2,x,0,0). If x < cT/2 then the curve is everywhere timelike, if x = cT/2 then the curve is everywhere null, and if x > cT/2 then the curve is everywhere spacelike. If you don't like the sharp bend in the curve then you can replace it with a family of helixes each of a different radius.
 

PAllen

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Consider a muon at rest at the origin of an inertial frame in flat spacetime. The worldline of the muon is a timelike curve connecting the creation and decay events. Let the lifetime of the muon be T, and consider the family of curves with 2 straight segments connecting the creation (0,0,0,0) and decay (cT,0,0,0) events through an event at (cT/2,x,0,0). If x < cT/2 then the curve is everywhere timelike, if x = cT/2 then the curve is everywhere null, and if x > cT/2 then the curve is everywhere spacelike. If you don't like the sharp bend in the curve then you can replace it with a family of helixes each of a different radius.
I am aware of this and even raised it in a different thread. Here I was making an unstated assumption: smooth paths, which would require that ds becomes spacelike when you replace the disconinuity in first deriviative with any smoothing. So I see only two minor qualifications to my overstatement:

1) Only one type of smooth pure path is possible between two given events; except

2) In a geometry allowing CTCs, there may be more than one type of path, each going through different parts of the geometry.

[In effect, the first is not really an exception because a non-smooth path has at least one point where ds is undefined, which, I can say, by definition is not a pure path]
 
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I am aware of this and even raised it in a different thread. Here I was making an unstated assumption: smooth paths, which would require that ds becomes spacelike when you replace the disconinuity in first deriviative with any smoothing. So I see only two minor qualifications to my overstatement:

1) Only one type of smooth pure path is possible between two given events; except
The helical paths I mentioned are smooth and can be purely timelike, spacelike, or null with no discontinuities.
 

PAllen

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The helical paths I mentioned are smooth and can be purely timelike, spacelike, or null with no discontinuities.
Can you describe this in more detail, either and equation or more detailed description. I don't see it at all, it seems impossible.
 

PAllen

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The helical paths I mentioned are smooth and can be purely timelike, spacelike, or null with no discontinuities.
Specifically, if this can be done at all, by scaling it should be possible to specify the following:

A smooth timelike curve in flat spacetime connecting the following points in Lorentz coordinates in some inertial frame: (t,x,y,z)=(0,0,0,0) and (0,1,0,0).

To be precise, by smooth I mean first derivative exists and is continuous everywhere.

I do not believe this is possible.
 
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PAllen

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Specifically, if this can be done at all, by scaling it should be possible to specify the following:

A smooth timelike curve in flat spacetime connecting the following points in Lorentz coordinates in some inertial frame: (t,x,y,z)=(0,0,0,0) and (0,1,0,0).

To be precise, by smooth I mean first derivative exists and is continuous everywhere.

I do not believe this is possible.
Note, if what Dalespam says is possible then a smooth timelike curve can connect events with spacelike separation. A smooth timelike curve is a physically possible worldline for a material particle. Thus Dalespam's claim is equivalent to saying a material particle can travel some distance in zero time (in some inertial frame).
 
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Can you describe this in more detail, either and equation or more detailed description. I don't see it at all, it seems impossible.
Consider the family of helical paths:
[tex]\left(ct,R \; cos(\omega t)+R, R \; sin(\omega t), 0\right)[/tex]
Where [tex]\omega=2\pi/T[/tex]

This helix connects the events (0,0,0,0) and (cT,0,0,0) with a smooth path. Those events are also connected by a straight timelike path.

The unnormalized tangent vector to the helix is:
[tex]\left(c,-R\omega \; sin(\omega t), R\omega \; cos(\omega t), 0\right)[/tex]

Which has a squared norm:
[tex]c^2-R^2\omega^2[/tex]
Which is timelike for
[tex]R(2\pi/T)<c[/tex]
Spacelike for
[tex]R(2\pi/T)>c[/tex]
And lightlike for
[tex]R(2\pi/T)=c[/tex]
 
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PAllen

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Consider the helical path:
[tex]\left(ct,R \; cos(\omega t), R \; sin(\omega t), 0\right)[/tex]
Where [tex]\omega=2\pi/T[/tex]

The unnormalized tangent vector is:
[tex]\left(c,-R\omega \; sin(\omega t), R\omega \; cos(\omega t), 0\right)[/tex]

Which has a squared norm:
[tex]c^2-R^2\omega^2[/tex]
Which is timelike for
[tex]R\omega<c[/tex]
Spacelike for
[tex]R\omega>c[/tex]
And lightlike for
[tex]R\omega=c[/tex]
But the timelike version will never connect points with spacelike separation.
 

PAllen

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This helix connects the events (0,0,0,0) and (cT,0,0,0) with a smooth path. Those events are also connected by a straight timelike path.
But, to replace the discontinuity of derivative in the so-called timelike path between spacelike events, you need to connect something like
(0,0,0,0) and (0,d,0,0). Any pair of points spanning the discontinuity can be put in the form in some inertial frame, in Lorentz coordinates.

That is the problem to be solved, and I claim it is impossible.

The GR extension simply amounts to the statement that whether two events are causally connected is invariant. If any two events can have a purely timelike path between them, then the lightcone structure of spacetime breaks down and all events are causally connected.
 

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