Timelike v. spacelike, is it arbitrary?

[WannabeNewton]

In fact that is exactly what you are doing when you go to a momentarily comoving locally inertial frame to an arbitrary timelike curve at some event on the curve.

 

[BruceW]

'when you go to a ... to a ...' the grammar does not work ?! But I think I understand what you mean.

Also, I was thinking about it again and I am not sure why they (on wikipedia) go on about the geodesics when they introduce normal coordinates. Any (not necessarily geodesic) smooth curve through the point can be expressed as $(tV^1,...,tV^4)$ so there's nothing interesting about the geodesics through the point is there? Maybe they make such a fuss about the geodesics because they can be uniquely associated with every tangent vector at the point? (there will be many different general curves which have the same tangent vector at the point). And on this wiki page http://en.wikipedia.org/wiki/Gauss's_lemma_(Riemannian_geometry) , they say that "any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point." But surely it is just as valid to say 'perpendicular to every smooth curve through the point'... it seems weird to me that they would specify geodesic for no particular reason.

 

[PeterDonis]

Maybe they make such a fuss about the geodesics because they can be uniquely associated with every tangent vector at the point?

Yes. Only geodesics have this property.

surely it is just as valid to say 'perpendicular to every smooth curve through the point'

No, because the point of the lemma is that geodesics radiating outward from the chosen point, when they cross a sphere of some small radius centered on the point, are perpendicular to the sphere, and that property is only true for geodesics. Non-geodesic curves will not all be perpendicular to the sphere, because they will "bend" relative to the geodesics. For example, think of curves radiating out from the north pole of a 2-sphere and crossing a curve of constant latitude some small distance south of the pole (the curve of constant latitude is the "sphere centered on the chosen point"--in this case a 1-sphere); the curves will all be perpendicular to the latitude line only if they are all geodesics, i.e., great circles.

Note that this lemma is for Riemannian geometry, not pseudo-Riemannian, so it is not directly applicable to spacetime.

 

[WannabeNewton]

'when you go to a ... to a ...' the grammar does not work ?! But I think I understand what you mean.

What I meant is that if $p$ is an event on some time-like curve $\gamma$ followed by a particle, then we can always find a locally inertial frame momentarily comoving with the particle at $p$ which is basically just the statement made above by Peter that at $p$, we can always find a time-like geodesic which intersects $\gamma$ and has the same tangent vector as $\gamma$.

 

[BruceW]

What I meant is that if $p$ is an event on some time-like curve $\gamma$ followed by a particle, then we can always find a locally inertial frame momentarily comoving with the particle at $p$ which is basically just the statement made above by Peter that at $p$, we can always find a time-like geodesic which intersects $\gamma$ and has the same tangent vector as $\gamma$.

ah right. yeah, that makes sense to me.

 

[BruceW]

No, because the point of the lemma is that geodesics radiating outward from the chosen point, when they cross a sphere of some small radius centered on the point, are perpendicular to the sphere, and that property is only true for geodesics. Non-geodesic curves will not all be perpendicular to the sphere, because they will "bend" relative to the geodesics.

that doesn't make sense to me. I thought we were saying that any non-geodesic curve close to the point can be locally identified with a geodesic through that point. In other words, that non-geodesic curves which go through the point can also be represented in normal coordinates as $(tV^1,...,tV^4)$. So, I thought that in the normal neighbourhood of the point, all curves are geodesic curves, in effect.

Note that this lemma is for Riemannian geometry, not pseudo-Riemannian, so it is not directly applicable to spacetime.

that's interesting. Is there some reason why it does not work for spacetime? Is it because we don't have the same notion of 'perpendicular' for pseudo-Riemmanian geometry?

 

[PeterDonis]

I thought we were saying that any non-geodesic curve close to the point can be locally identified with a geodesic through that point.

In the sense that they both have the same tangent vector at the point, yes. But only at the point. See below.

In other words, that non-geodesic curves which go through the point can also be represented in normal coordinates as $(tV^1,...,tV^4)$.

No; only geodesics will appear as straight lines in normal coordinates throughout the entire coordinate patch surrounding the point. Non-geodesic curves will appear as curved lines that are tangent to the geodesic straight lines at the chosen point.

Is there some reason why it does not work for spacetime?

Yes; you can't define a "sphere" the same way, because the metric is not positive definite. There might be theorems which are somewhat analogous for spacetime, but I don't know of any off the top of my head.

 

[BruceW]

In the sense that they both have the same tangent vector at the point, yes. But only at the point. See below.
...
No; only geodesics will appear as straight lines in normal coordinates throughout the entire coordinate patch surrounding the point. Non-geodesic curves will appear as curved lines that are tangent to the geodesic straight lines at the chosen point.

aahh, ok, I get it now. I suppose that is the entire point of the exponential map in the first place. Thanks very much for explaining all this to me. I appreciate it. And I have one last question, well, really I'm not totally sure of why they talk about a normal neighbourhood of the point, within which the geodesics are in the form $(tV^1,...,tV^4)$
I mean, what I expect is that in some kind of limit of 'closeness' to the point, the geodesics would tend to the form $(tV^1,...,tV^4)$ But they (wikipedia) explain it in a different way. They use this idea of a normal neighbourhood, and don't mention limits as far as I can tell.

 

[WannabeNewton]

Neighborhoods encode all the information about limits in topological spaces. Keep in mind that an arbitrary space-time manifold has no natural metric so $\epsilon$-$\delta$ definitions of limits don't apply. In topological spaces, neighborhoods fully characterize limits; for example a sequence $(x_i)_{i\in \mathbb{N}}$ in a topological space $X$ converges to a point $p\in X$ iff for any neighborhood $U$ of $p$, there exists an $N\in \mathbb{N}$ such that for all $n > N$, $x_n \in U$. This is quite similar in form to the definition of the convergence of a sequence of points in a metric space except that there is no metric when dealing with arbitrary topological spaces hence no notion of "closeness" in the metric sense (of course for a metric space the two definitions are equivalent); this is for example why you don't naturally see Cauchy sequences defined for arbitrary topological spaces. You would probably gain a better understanding of these things by shifting through a proper Riemannian geometry text e.g. Lee "Riemannian Manifolds: An Introduction to Curvature" or O'Neill "Semi-Riemannian Geometry With Applications to Relativity".

 

[PeterDonis]

I suppose that is the entire point of the exponential map in the first place.

Yes, exactly.

 

[BruceW]

Neighborhoods encode all the information about limits in topological spaces. Keep in mind that an arbitrary space-time manifold has no natural metric so $\epsilon$-$\delta$ definitions of limits don't apply. In topological spaces, neighborhoods fully characterize limits...

ah great, I'm glad there is meaning behind it. and thanks for the names of texts.

 

[ghwellsjr]

First, a clarification: strictly speaking, the terms timelike, spacelike, and null apply to *vectors*, not curves--more precisely, they apply to tangent vectors to curves at particular points.

Could you also please provide a definition of "spacetime interval" as applied to Special Relativity?

Thanks.

 

[ghwellsjr]

Indeed worldline refers only to time-like curves in space-time.

As I said in post #133, Taylor and Wheeler also include lightlike paths.

Who's right? Where do we go to get the commonly accepted definitions for words like "worldline"?

 

[robphy]

As I said in post #133, Taylor and Wheeler also include lightlike paths.

Who's right? Where do we go to get the commonly accepted definitions for words like "worldline"?

The restriction of "worldline" to timelike curves is not universal. From a google text search, Synge, Sachs&Wu, Schutz, and Ludvigsen also apply worldline (or "world-line" or "world line") to light. There are certainly others.

So, such a term is likely defined locally within the text or article. If precision is needed, one should use a more descriptive phrase like "timelike curve", which also should be defined or refined if needed.

 

[ghwellsjr]

The restriction of "worldline" to timelike curves is not universal. From a google text search, Synge, Sachs&Wu, Schutz, and Ludvigsen also apply worldline (or "world-line" or "world line") to light. There are certainly others.

So, such a term is likely defined locally within the text or article. If precision is needed, one should use a more descriptive phrase like "timelike curve", which also should be defined or refined if needed.

But:

First, a clarification: strictly speaking, the terms timelike, spacelike, and null apply to *vectors*, not curves--more precisely, they apply to tangent vectors to curves at particular points.

For the lightlike case, the "curve" is null having a "length of zero" and therefore I would think the expression "tangent vectors to curves at particular points" would have no meaning. Is this the reason some authorities exclude "lightlike" from the definition of "worlineline", simply because there is no line there?

Also, do all authorities agree that "worldline" cannot apply to a spacelike curve? For example, is definition.com at least correct that "the path of a particle in space-time" excludes spacelike curves? If so, is there a more technical term than "path" or "curve" that includes all three of the different categories (timelike, lightlike or null, and spacelike) so that we can talk about the interval between two events without knowing or specifying which one it might be?

 

[BruceW]

(I think) you can still find the tangent vector at points along a null curve. It is just that when you integrate, you get zero. And that's a good question about a general term for timelike, null and spacelike curves. I've seen simply the word 'curves' used. But I'm not sure if this is standard.

 

[PeterDonis]

For the lightlike case, the "curve" is null having a "length of zero" and therefore I would think the expression "tangent vectors to curves at particular points" would have no meaning.

No, a null curve can still have a perfectly well-defined tangent vector at each of its points, it's just that that vector will have zero length according to the metric. Tangent spaces and vectors within them exist independently of the metric that assigns a zero length to certain vectors.

is there a more technical term than "path" or "curve" that includes all three of the different categories (timelike, lightlike or null, and spacelike) so that we can talk about the interval between two events without knowing or specifying which one it might be?

I've never seen any other general term besides "curve" that includes all three categories. (The term "geodesic" can be applied to a curve in any of the three categories, but of course not all curves are geodesics.)

 

[robphy]

Since Euclidean/Riemannian geometry has a positive-definite metric,
there is essentially only one kind of vector.
However, for pseudoRiemannian geometry, there are more than one type:
for Lorentzian-type.. three kinds (timelike, spacelike, null) ;
for Galilean-type, two kinds (spacelike and null coincide).

In the pseudo-Riemmannian cases, I would use "curve" to be the most general... including zigzaggy ones (Penrose's "bad trip") or ones that may change the type of tangent-vector, which is mathematically allowed. To less-ambiguously refer to specific subsets of curves, one should use additional terms, e.g. [smooth] future-timelike curve. (Some references [like Penrose's "Techniques of Differential Topology in Relativity"] distinguish between paths (a mapping) and curves (the image of a mapping).)

 

[WannabeNewton]

A curve is simply a map from an interval in the reals into the smooth manifold such that the map itself is smooth. The qualifiers timelike, nulllike, or spacelike for curves just means that the tangent vector field to the curve is of that character.

 

[ghwellsjr]

However, there is a commonly accepted standard definition for spacetime interval in SR and it excludes curved paths. It is always the longest ... path between any two arbitrary events.

Where do you get that definition of spacetime interval from? I'm pretty sure that is not the standard definition. For example, in the twin paradox, we talk about the spacetime interval along the path taken by the accelerating twin. (and since the twin is accelerating, this is not the longest worldline between events).

Every book I have seen, every article I have seen, every definition I have seen states that the spacetime interval between two events is invariant so we never talk about it with reference to the path taken by the accelerating twin.

Can you point to one reference that says that the spacetime interval applies equally to both twins?

 

[Dale]

Every book I have seen, every article I have seen, every definition I have seen states that the spacetime interval between two events is invariant so we never talk about it with reference to the path taken by the accelerating twin.

Can you point to one reference that says that the spacetime interval applies equally to both twins?

It is the same thing. If you can measure the distance between two close points (ds) then you can integrate those distances to get the total distance along a path (s).

Things get more complicated in curved spaces. In flat spaces there is one unique path which is the extremal distance, and so you can say that the length of that path (s) is the distance between the points. However, in curved spaces there may be multiple extremal paths, so it is difficult to call the length of one the distance.

For a reference I would read Carroll's lecture notes on GR, but it isn't a single part of the notes but rather a synthesis of the first few chapters.

 

[ghwellsjr]

It is the same thing. If you can measure the distance between two close points (ds) then you can integrate those distances to get the total distance along a path (s).

Things get more complicated in curved spaces. In flat spaces there is one unique path which is the extremal distance, and so you can say that the length of that path (s) is the distance between the points. However, in curved spaces there may be multiple extremal paths, so it is difficult to call the length of one the distance.

For a reference I would read Carroll's lecture notes on GR, but it isn't a single part of the notes but rather a synthesis of the first few chapters.

I was specifically talking only about Special Relativity. And the question is can the Spacetime Interval apply to both the accelerating twin and the inertial twin?

 

[Dale]

Yes. If you can measure the "straight" interval between two close points (ds) then you can integrate those intervals along a curved path to get the total interval along the path (s).

 

[PAllen]

I think there is a tendency to use spacetime interval both for something that is a function of two events (in which case you are implicitly discussing either SR or a sufficiently small region of spacetime in GR, such that the geodesic is unique), or as a function of a path in spacetime. It would be nice if there were clearer terminology, but generally the context is used to disambiguate (are we talking only about points/events, or are one or more paths being considered).

 

[robphy]

I think there is a tendency to use spacetime interval both for something that is a function of two events (in which case you are implicitly discussing either SR or a sufficiently small region of spacetime in GR, such that the geodesic is unique), or as a function of a path in spacetime. It would be nice if there were clearer terminology, but generally the context is used to disambiguate (are we talking only about points/events, or are one or more paths being considered).

I think of the spacetime interval [between two nearby events] as [the square of] an infinitesimal bit of arc-length (the line element), dependent on the event and an infinitesimally-close nearby event.

For the case of a timelike curve, the integrated arc-length [along that path] is the proper-time (measured by a clock that traveled along that curve).

These notions are distinct.

Because of the symmetry properties of SR,
along a geodesic ("straight line") in SR,
the two notions can be loosely identified.

I would not, however, loosely identify the spacetime-interval between two events with
the proper-time along the traveling twin's non-geodesic worldline between those events.

[edit...]
p.s.
The spacetime-interval between two events is more closely related to an infinitesimal "displacement",
whereas the integrated proper-time along an arbitrary path between is closely related to the "total distance traveled".

 

[PAllen]

I think of the spacetime interval [between two nearby events] as [the square of] an infinitesimal bit of arc-length (the line element), dependent on the event and an infinitesimally-close nearby event.

For the case of a timelike curve, the integrated arc-length [along that path] is the proper-time (measured by a clock that traveled along that curve).

These notions are distinct.

Because of the symmetry properties of SR,
along a geodesic ("straight line") in SR,
the two notions can be loosely identified.

I would not, however, loosely identify the spacetime-interval between two events with
the proper-time along the traveling twin's non-geodesic worldline between those events.

[edit...]
p.s.
The spacetime-interval between two events is more closely related to an infinitesimal "displacement",
whereas the integrated proper-time along an arbitrary path between is closely related to the "total distance traveled".

But, especially in SR, one speaks of an interval between points with finite separation, where the interval can be of any character. Even in GR, this is possible within a 4-volume such that the geodesic is unique. (This is the basis if Synge's world function of pairs of events with finite (but not too large) separation, with the interval being of any character). With interval specified between points with finite separation, you must implicitly assume a unique geodesic between them.

Similarly, integrated interval can be applied to any path all of whose tangents are of the the same character (or, with trickery I find devoid of meaning, to any path at all). The integrated interval along a path will be a proper time if all tangents are causal, and a proper length if all are a-causal. The proper time has a much more direct physical meaning, but the length can also be given a physical meaning:

- each nearby pair of points is such that there is local observer for whom they are at rest [edit: simultaneous is really what is important], and who can measure a ruler distance between them. Add these up over the curve, and you get its proper length (also taking the limit to zero of affine parameter between successive points).

 

[WannabeNewton]

The restriction to sufficiently small open sets is indeed very essential with regards to uniqueness and maximality of length for time-like geodesics. It is not true that between any two points there is a unique time-like geodesic that maximizes the proper time. For example, one can take Minkowski space-time and roll it up into a cylinder; one can find two points between which there are two time-like geodesics (one that goes straight up the cylinder and another which coils around it) and it is the latter which maximizes the proper time.

 

[PeterDonis]

one can find two points between which there are two time-like geodesics (one that goes straight up the cylinder and another which coils around it) and it is the latter which maximizes the proper time.

Don't you mean the former? The one that goes "straight up the cylinder" (i.e., the one with a winding number of zero) is the one that globally maximizes the proper time. (I say "globally" because each geodesic maximizes the proper time locally--in this case, within the class of all curves with the same winding number.)

 

[WannabeNewton]

Oops sorry, yes I meant the former! My apologies.

 

[BruceW]

that is awesome. Is that because 'flat' and 'cylinder' are homeomorphic? Or is it diffeomorphic? (I don't know much about isomorphisms, but I think they are pretty interesting).

 

[WannabeNewton]

Well let's take the unit square in 1+1 Minkowski space-time. To roll up the unit square into a cylinder, we simply curl every horizontal strip of the square into a circle i.e. we identify the endpoints of each strip. So on the plane, if we wanted to connect say $p = (0,\frac{1}{4})$ to $q = (0,\frac{1}{2})$ using a line, there is only one such line and it is simply the line $l_1$ drawn straight up from $p$ to $q$. We can also draw a line $l_2$ from $p' = (1,\frac{1}{4})$ to $q$; note that both lines are time-like curves. On the plane these lines of course start at different points but on the cylinder these points are identified so the two lines get mapped onto curves that have the same starting and ending points on the cylinder. The cylinder and plane are locally isometric so the geodesics of the plane (the lines) get mapped onto geodesics of the cylinder and arc-length will be preserved, meaning that the images of $l_1,l_2$ will be geodesics on the cylinder and that the image of $l_1$ will have greater length than the image of $l_2$ on the cylinder.

 

[BruceW]

ah right, we want local isometry, since we want the 'same' metric on both. And (I think) we also need local diffeomorphism to get local isometry. And the reason we can only get a local diffeomorphism, but not a diffeomorphism is because of the 'edges' of the manifold?

 

[WannabeNewton]

Local isometry implies local diffeomorphism. We can't have a global diffeomorphism or even a global homeomorphism because the cylinder doesn't have a trivial fundamental group like the plane does i.e. the plane is simply connected but the cylinder is not (the fundamental group of the finite cylinder is infinite cyclic), and being simply connected is a topological invariant.

 

[BruceW]

Local isometry implies local diffeomorphism.

wikipedia seems to imply differently. http://en.wikipedia.org/wiki/Isometry_(Riemannian_geometry)
But on the other hand, they say here http://en.wikipedia.org/wiki/Diffeomorphism that "It is easy to find a homeomorphism that is not a diffeomorphism, but it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2, 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs have been found. The first such example was constructed by John Milnor in dimension 7." So I guess this means if we have a local isometry then we are likely to have local diffeomorphism, but is not certain?

We can't have a global diffeomorphism or even a global homeomorphism because the cylinder doesn't have a trivial fundamental group like the plane does i.e. the plane is simply connected but the cylinder is not (the fundamental group of the finite cylinder is infinite cyclic), and being simply connected is a topological invariant.

ah, that makes sense. cool.

 

[WannabeNewton]

A local isometry is by definition a local diffeomorphism such that the metric tensor is preserved under pullback. So a map being a local isometry immediately implies it is a local diffeomorphism by definition.