# Timelike tangent vectors in Zwiebach

1. Oct 26, 2004

### Jimmy Snyder

On page 98 of Zwiebach's book "A First Course in String Theory", the following claim is made:

At each point on the worldsheet of a string there is both a spacelike and a timelike tangent vector.

Professor Zwiebach acknowledges that the statement needs to be softened as follows:

At each point on the worldsheet, except those points associated with the endpoints of an open string, there is both a spacelike and a timelike tangent vector. At the points associated with endpoints, there is both a spacelike and a null tangent vector.

I don't understand the proof that he gives. Can someone provide me with an alternative proof, or a clearer version of his proof? Or is the theorem perhaps untrue? Are there interior points with a null tangent vector, but no timelike tangent vector?

2. Oct 26, 2004

Staff Emeritus
Isn't this the same question you had on your thread String Question? why did you start a new one? Mentors don't like double posting, you know.

3. Oct 26, 2004

### Jimmy Snyder

4. Oct 26, 2004

Staff Emeritus
Ah, OK. Then let me know if you understand the two paragraphs near the bottom of page 98:
The one beginning "To appreciate the need..."
and the one beginnig "The argument for the string...".

The first of these paragraphs supplies an argument ( a little less that a proof in all rigor, I suppose) for the existence of a timelike vector at each point of the worldline of a particle. The argument is by contradiction; he supposes the opposite of what he wants to prove and shows that leads to "an unphysical situation". This is from the contrapositive mode of logic; if not-A is false then A is true. Are we agreed on that first paragraph and on the way its argument proceeds?

5. Oct 26, 2004

### nrqed

Hi Jimmy,

your question forced me to think much more about this whole issue and I think I understand much better what is going on (anyway, that's how I am feeling right now at midnight, after having been at school for the last 17 hours so my judgment might not be top notch )

I really think now that we must think of a massive string, so no actual piece of the string may move at c. All pieces *must* move at a speed below c. It is still true that we can not follow individual little sections of the string, but I think that for now we can imagine that we could, and then the statement is that no piece would move above or even at c. Then it's clear that at each point of the worldsheet there is a timelike tangent vector. Because (again, imagining for now that we could follow a small piece), there must be some frame in which this small piece is at rest.

Now the interesting thing is the end points. What I failed to fully appreciate before is that the endpoints are NOT physical pieces of the string. They are just the locations of the extremities of the string. If we would be considering actual real life strings, the end points would be the extremities of the last atoms in the string. So even though the end points move at c, there is no actual matter moving at c. What is really amazing of course is the fact that the endpoints must move at c, It is really a fascinating result. But, again, all the actual pieces of "stuff" making up the string are moving more slowly than c. This implies that the string cannot be in translation (the endpoints would move at c and the main body of the string would move more slowly, so the string would get longer and longer!). A possible motion would be for the string to be spinning (let's say around its midpoint). Then the endpoints can move at c while all the pieces move more slowly than c.

Do you see what I mean? Thus point of view makes the discussion of Zwiebach more natural because he compares the string to a point particle which is massive. And indeed, the body of the string always moves below c, like the point particle. The truly fascinating result is that the end points of the free string *must* move at c! I can follow the maths leading to this but it is a bit counterintuitive. I don't see any clear *physical* argument!

Regards

Pat

6. Oct 27, 2004

### Jimmy Snyder

Yes. Of the two paragraphs you mention, the first is no problem for me, and the second one is the one for which I would appreciate any insight. Thank you for picking up this thread.

7. Oct 27, 2004

Staff Emeritus
OK. My thinking is that the first two sentences of the second paragraph, about the radical nondecomposability of the string, should be treated as a side issue and not hold us up, since he says "as we shall make abundantly clear later on".

Then the contrapositive argument on the worldsheet (with an exception for the end points of the open string) just parallels that of the first paragraph, on the worldline of a particle. Do you have any particular objection to this argument? Do you think it is just hand-waving? If so why wasn't the particle argument also hand-waving?

8. Oct 27, 2004

### nrqed

The obvious question is why an exception for the end points? What is the argument that says that we must make those points an exception? From the way I see it, this is the crux of the issue. (I already wrote in a previous post how I no wunderstand it, but I might be wrong).

Pat

9. Oct 27, 2004

Staff Emeritus
Pat, I want to put off these general questions until we get resolution on whether Zwiebach's contrapositive argument about all possible Lorentz observers is both valid and strong enough to convince us.

10. Oct 27, 2004

### Jimmy Snyder

Agreed

I'm not sure if I can agree with you on this. However, you do provide much food for thought.

We are trying to prove that "at each point on the worldsheet there is a timelike tangent vector". To assert the contrapositive he should have "Suppose there is a point on the worldsheet where there is no timelike tangent vector". I do not see an equivalent to this statement in his proof.

By analyzing the next few sentences in the manner you suggest, I see that we could break up his proof into three pieces like this:

Lemma 1: Among the points along a closed string, there is at least one point for which there is a timelike tangent vector to the worldsheet.

Lemma 2: Among the points along a piece of a closed string, there is at least one point for which there is a timelike tangent vector to the worldsheet.

Theorem: For each point along a closed string, there is a timelike tangent vector to the worldsheet. The statement of this last corollary is missing from the text, but should be placed just before the sentence which begins "Since the endpoints of the rest of the string ..."

Or so it seems to me. Is this the structure of the proof?

11. Oct 27, 2004

### Jimmy Snyder

You are right, this is the crux of the issue. selfAdjoint is generously applying himself to this problem and we should give him some leeway here. However, eventually we will have to revisit this question.

Here is where it stands for me.

1. I don't understand the proof. selfAdjoint is addessing this issue and I am pleased with the results so far.

2. I don't trust that the proof is correct. You have stated the reason why. We will tackle this issue once the first one is done. However, it may be that if I understood the proof, I would see why this was not an issue.

3. I don't believe the theorem. I wonder if the statement is not a consequence of first principles, but rather is itself a first principle. In other words, instead of a theorem with proof, it should be replaced with the words:

For the remainder of this book, we will assume that at each point on the worldsheet, there is a timelike tangent vector.

Again, if I understood, and trusted the proof, then my third concern would melt away.

12. Oct 27, 2004

Staff Emeritus
Is the reason you don't understand the proof the question of the endpoints? If that is so then let's discuss the endpoints. All the physics of the string is done with the worldsheet, which is treated as a two dimensional manifold, a curved surface. But in the case of the open string it's a manifold with boundary, because the world lines of the endpoints form edges of the worldsheet. Thus these worldlines are distinguished on the world sheet. While the timelike and spacelike vectors on the interior of the worldsheet have to be established, the edges are just there, independent of any coordinate system or anything. The geometry you establish on the worldsheet has exceptions for the edges, you have to do a little extra dido there; this isn't a shortcoming or a problem, it's just a geometrical fact, which the math is well able to deal with. In fact in Polchinski's book it's one of the first excercises.

I do suggest we pretend we are working with a closed string, so the endpoint issue doesn't come up, and see how the rest of the proof goes there.

Last edited: Oct 27, 2004
13. Oct 27, 2004

### nrqed

Hi Jimmy,

I don't know if you have read my post from yesterday night but I have adopted a point of view which makes, in my personal opinion, the presentation of Zwiebach completely satisfactory (to my taste, at least). SO I am quite happy with the situation, now. I'll paste part of that previous post I had below. It might be that it's not a satisfactory point of view for you in which case I sure would benefit from your criticisms. In any case, I appreciate your posts because it forced me to think about the whole issue and not to let myself set it aside as I did on my first reading. And I now feel I understand much better that aspect of string behavior.

Anyway, here's part of what I wrote yesterday:

Pat

14. Oct 27, 2004

### Jimmy Snyder

No, not at all. It is because of the question of the endpoints that I don't trust the proof. Here is the issue: The statement of the theorem doesn't mention endpoints and the proof itself doesn't mention endpoints. Why then doesn't the theorem apply to endpoints? But please, let us leave that question till later.

The reason I don't understand the proof is that I can't follow it. To my feeble lights, it seems to have missing steps. What I am looking for is for someone to either provide the missing steps, or perhaps to provide a completely different proof. For instance, is there a different proof in Polchinsky?

Yes, by all means.

15. Oct 27, 2004

Staff Emeritus
Great, so we pick up Zwiebach's argument in the third sentence of that second paragraph: (my comments are in red)

"For a closed string world-sheet, for example, consider first the possibility that that all along the string (all around the closed loop, that is) there is no timelike tanget vector to the world sheet. (Here he is assuming the contradiction of what he wants to prove, and his proof will be to show that contradictory assumption is false, or in physical terms "unphysical").

Then he just asserts the contradiction:

That means we could display all possible Lorentz observers at all points on the string and no observer could make any point of the string appear to be at rest. (Because if one observer saw one point at rest then that rest frame, be definition would have a timelike vector at that point)

And extends it:

A similar unphysical result would occur if any piece of the (closed) string failed to have timelike tangent vectors on the world-sheet. (This is the real contradiction, and the next sentence is the proof of it. A piece is a length of string from one interior point to another one.)

Since the endpoints of the rest of the string can not close up the string instantaneously, a piece of the string would have failed to move physically.

(Note that if a piece moves, time must pass, and that piece must have a timelike tangent somewhere along it. The only way to move without time passing is to have a null tangent vector, i,e, to move at the speed of light. But the interior of the string doesn't do that. Failing that, the only alternative is for the piece of the string to disappear, but without time passing (timelike tangent), that would have to happen instantaneously, and for points (the end points of the PIECE; interior points of the STRING) to move a finite distance (length of the piece) in no time is unphysical again. So the whole idea is unphysical, so the assumption is false, and there does exist a timelike tangent somewhere along the piece. Now go to many increasingly small pieces and you eventually get timelike tangents at a dense set of points on the string, and using continuity claim that every point on a closed string has one.)

Last edited: Oct 27, 2004
16. Oct 27, 2004

### Jimmy Snyder

Hi Pat,

I did read your post, and I had intended to respond to it, but just completely forgot to do so. I appreciate your input at least as much as you appreciate mine. I feel very strongly that if we all pull together, we can lick this thing.

Consider a string that is circular, not vibrating, and at rest wrt an inertial observer. The world sheet is a right cylinder. Quite obviously, there are timelike tangent vectors everywhere. Now 'paint' a point of the string as you suggested, and (momentarily suspending physical reality) follow the worldpath of the point. If the the string is stationary in the sense that the point is also at rest wrt to the observer, the worldpath is a straight line parallel to the axis of the cylinder. If the string is rotating in the sense that the point is moving in a circular path coincident with the string, then the world path is a spiral winding around the cylinder. Note that in either case, the world sheet is the same. In other words, the world path of the point has no effect on the shape of the worldsheet and therefore no effect on the question of whether there are timelike tangent vectors. This is true even if the string is rotating in such a way that our painted point is traveling at or even greater than c. The issue is the shape of the world sheet, not the path of any point. I expect the proof to rely on the geometry and perhaps the topology of the worldsheet. Indeed, I think there is some topology in Zwiebach's proof since he mentions not being able to patch up some gap.

All this is to say that I don't like to think of being able to follow any part of the string.

There is a very important difference between the endpoints of a string theory string, and the endpoints of a macroscopic string. If you cut off the endpoints of a macroscopic string, it still has endpoints, albeit different ones. But if you cut off the endpoints of a string theory string, it becomes a string with no endpoints. That is why I was so interested in selfAdjoint's suggestion that the endpoints are constituents of the string. You could actually cut them off and consider them as separate entities. Then the world would be made of strings and endpoints. Obviously, a statement like that must give us pause. Everything is supposed to be made of strings. I don't know how to resolve this issue. If a string has no endpoints, then it has nothing to attach it to branes. There is no point to apply Dirichlet or Neuman boundary conditions. Perhaps the answer is that you cannot cut off the endpoints. But then your objection comes to the fore. If the string is traveling in a straight line, and the endpoints are traveling at c and the interior points are traveling at less than c, then the string must get stretched in a bizarre way.

Too cloudy here to see the eclipsed moon. Second time in a row. Disappointment abounds.

17. Oct 27, 2004

### Jimmy Snyder

18. Oct 28, 2004

Staff Emeritus
I had my comments in red. They display on my sceen (IE), but here, I'll change them to italics.

----------------------------------------------------------------------
Great, so we pick up Zwiebach's argument in the third sentence of that second paragraph: (my comments are in italics)

"For a closed string world-sheet, for example, consider first the possibility that that all along the string (all around the closed loop, that is) there is no timelike tanget vector to the world sheet. (Here he is assuming the contradiction of what he wants to prove, and his proof will be to show that contradictory assumption is false, or in physical terms "unphysical").

Then he just asserts the contradiction:

That means we could display all possible Lorentz observers at all points on the string and no observer could make any point of the string appear to be at rest. (Because if one observer saw one point at rest then that rest frame, be definition would have a timelike vector at that point)

And extends it:

A similar unphysical result would occur if any piece of the (closed) string failed to have timelike tangent vectors on the world-sheet. (This is the real contradiction, and the next sentence is the proof of it. A piece is a length of string from one interior point to another one.)

Since the endpoints of the rest of the string can not close up the string instantaneously, a piece of the string would have failed to move physically.

(Note that if a piece moves, time must pass, and that piece must have a timelike tangent somewhere along it. The only way to move without time passing is to have a null tangent vector, i,e, to move at the speed of light. But the interior of the string doesn't do that. Failing that, the only alternative is for the piece of the string to disappear, but without time passing (timelike tangent), that would have to happen instantaneously, and for points (the end points of the PIECE; interior points of the STRING) to move a finite distance (length of the piece) in no time is unphysical again. So the whole idea is unphysical, so the assumption is false, and there does exist a timelike tangent somewhere along the piece. Now go to many increasingly small pieces and you eventually get timelike tangents at a dense set of points on the string, and using continuity claim that every point on a closed string has one.)

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19. Oct 28, 2004

### nrqed

Hi Jimmy,

Thanks for your reply and very interesting comments. I agree with all that you are saying. Indeedd, I can't really follow any of the individual pieces of the string because there is no way to identify them, as you point out. And indeed, if a closed string is at rest completely or spinning around the center of its axis can't be distinguished.

But it's possible to define a velocity by using the method described by Zwiebach on page 109. I quote :

"consider a string at some fixed time and pick a point p on it. Draw the hyperplane orthogonal to the string at p. An infinitesimal instant later the string has moved, but it will still intersect the plane, this time at a point p'. The transverse velocity is what we get if we presume that the point p moved to p'. No string parameterization is needed to define this velocity"

Then my statement is that I think that the key point is that this velocity is taken to be smaller than c, for any point on the string. Given that, the rest of the discussion by Zwiebach on p. 98 makes sense.

I have to say that I totally disagree with this. Of course I could be wrong, but I just can't make sense of it. The endpoints are just defined at the points where the string ends. Even if the string is a fundamental string, it does not change the meaning of what an endpoint is. If you cut an open string in two, you end up with two open strings having each two end points. If you "cut an endpoint" of an open string, you are cutting a small piece of the string and are creating two open strings (however short one may be compared to the other).

I really think that endpoints here are just the usual concepts of endpoints!!

The subtle point of treating a fundamental string, as opposed to a real life macroscopic string, is the point you raised about not being able to follow individual pieces of the string in their motion. Then it becomes impossible to really say if they are moving above or below c, or at c. Then it makes the entire analogy with a point particle quite tricky.

My contention is that the equivalent of saying that the point particle moves below c is equivalent to saying thatnone of the transverse velocities at any point along the string (as defined above) moves above or even at c. Then the rest would follow. However, I agree that this is not the way Zwiebach presents things, so I don't know if I am correct. MAybe he wanted to discussed the presence of timelike tangent vectors before getting into the exact definition of transverse velocities and he therefore had to cheat a little bit.

But again, maybe I am totally wrong. To be honest then, if we don't impose the transverse velocities to be below c and since we can't follow any individual piece of the string, I would not see anyway to make any statement about timelike vs null vs spacelike tangent vectors! But given the result (which I trust is correct), then the only explanation thatmakes sense to me is the one I just presented.

Right. So if I am right, then the string *must* be spinning. That way it's always possible to have the endpoints moving at c while all the transverse velocities remaining below c. That's quite an intriguing conclusion! It seems to me that this should be mentioned somewhere, but I have never seen (or recall seeing) anywhere the mention that the classical string must be spinning!

Too bad. It was quite clear here. Got quite reddish. It's easy to imagine how it must have provoked awe for ancient civilizations.

Pat

20. Oct 28, 2004

### nrqed

But that's the crux of the matter. How do we know that the interior of the string can't do that? That's the key assumption. And that was my starting point in a much earlier post in this thread.

But *even* if we accept this statement, it is still unclear what one means by "the interior can't move at the speed of light" since we can't follow an individual piece of the string. The only possible answer I came up with was to work in terms of transverse velocities, as defined on page 109. See my earlier post today for my discussion on this.

regards

Pat